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From patterns to generalizations: sequences and series

Skills check 1 a

3x  5x  20  20x  4

 8x  20x  20  4  12x  24 x  2

x 1 x 3  2x  1 2x  1

  x  1 2x  1   x  3 2x  1  2 x 2  3x  1  2 x 2  7 x  3  10x  2 x 

1  2   2 1  2 1  2  2

12 2 2 32 2  12 1

x 1 2   x  1 2x  1 x  1

x 2x  1  x  1   x  1  x  1  2  x  1 2 x  1

 x  1 2x  1  x  1

x 2 x  3x  1  x 2  1  2 2 x 2  x  1

2 x  3x  x  x  1  4 x 2  2 x  2

2 x 3  3x  1 (x 2  1)(2x  1)

Exercise 1A 1 a

Next three terms are 9, 10.5, 12 The sequence is obtained by adding 1.5 to the previous term and can be written as 3, 3  1.5, 3  2(1.5), . 3  (n  1)(1.5) 

b Next three terms are 5, 2, -1 The sequence is obtained by subtracting 3 from the previous term and can be written as 17, 17   3 , 17  2(3), . 17  (n  1)(3) 

un  20  3n, n  c

Next three terms are 243, 729, 2187 The sequence is obtained by multiplying the previous term by 3 and can be written as 3, 3  3, 3  32, 3  33, . 3  3n1

un  3n d Next three terms are

13 16 19 , , 16 19 22

The sequence is obtained by adding 3 to both the previous numerator and denominator and 1   n  1 3 1 1  3 1  2  3  1  3 3  , , , . can be written as , 4 4  3 4  2  3  4  3 3  4   n  1 3 un 

e Next three terms are

1 1 1 , , 90 132 182

The sequence can be written as

1 1 1 1 , , , . 12 3 4 5 6 (2n  1)(2n)

ur  3  2r u1  3  2  1 u2  3  2  2  1 u3  3  2  3  3 u4  3  2  4  5 u5  3  2  5  7

1 2 3 4 5 , u2  , u3  , u4  , u5  2 1  1 22 1 23 1 24 1 25 1

1 2 3 4 5 , , , , 3 5 7 9 11

u1  2  1   1  1  1 1

u2  2  2   1  2  6 2

u3  2  3   1  3  3 3

u4  2  4   1  4  12 4

u5  2  5   1  5  5 5

u1  u2  u3  u4  u5  3,

3 211 3 22 1 3 23 1 3 24 1 3 25 1

3 3 2 3  4 3  8 3  16 

3 3 3 3 , , , 2 4 8 16

3 a 5, 10, 15, 20, …. The multiples of 5

b 6, 14, 22, 30, … The sequence is obtained by adding 8 to the previous term and can be written as

c The sequence is obtained by multiplying the previous term by

1 and can be written as 2

d The sequence is obtained by multiplying the previous term by 

1 and can be written as 3

e The sequence can be written as 0  2, 1  3, 2  4, 3  5, . (n  1)  (n  1) OR The sequence can be written as 12  1, 22  1, 32  1, 42  1, .

ur   r 2  1 , 4 a

 2r 1  r   0  4  12  24 r 1

r 2  0  1  4  9  16  25

 3r  1  2  5  8  11  14 r 1

Explanation: think of this as 4

r 1 11 2 1 3 1 4 1 5 1  2  2  2   2  . 2 1 2 3 42 5 r 1 r 

3 4 5 6     . 4 9 16 25

 1   1   1   1   1  . 2 2 2 2 2 2 1  1 2 2  1 2 3  1 2  4   1 2 5  1 1

1 1 1 1     . 7 17 31 49

 r 5r  1  1 5  1  1  2 5  2  1  3 5  3  1  4 5  4  1  5 5  5  1  . r 1

 4  18  42  76  120

 3  20  3  21  3  22  3  23  3  24  3  .

 11  22  33  44  55  .  1  4  27  256  3125  .

6 a The series can be written as 8  8  3  8  2  3  8  3  3  8  4  3 It has five terms and the general term can be written as ur  11  3r 5

b The series can be written as 1  3  2  5  3  7   4  9  5  11 It has five terms and the general term can be written as ur  r  2r  1 5

c The series can be written as

0 1 2 3 4 5       . 2 3 4 5 6 7

It is an infinite series and the general term can be written as ur  6

d The series can be written as 1²  3²  5²  7²  9² It has five terms and the general term can be written as ur  2r  1 ² 5

e The series consists of the multiples of 3k It has five terms and the general term can be written as ur  r  3k  5

Exercise 1B 1 a

u1  3, d  5  un  3  5  n  1  5n  2

u1  101, d  4  un  101  4  n  1  105  4n

u1  a  3, d  4  un  a  3  4  n  1  4n  a  7

 un  20  15  n  1  15n  35

u1  5, d  6  u15  5  6 15  1  5  6 14  89

u1  10, d  7 u11  10  7 11  1  10  7 10  60

u1  a, d  2 u17  a  2 17  1  a  2 16  a  32

u1  16, d  4  un1  16  4  n  1  1  16  4n

u1  16, d  5 un  21  5n  64  5n  85  n  17

u1  108, d  7 un  7n  115  60  7n  175  n  25

u1  15, d  4 un  11  4n  95  4n  84  n  21

u1  2a  5, d  2 un  2n  2a  7  2a  23  2n  30  n  15

u1  5 1  7  2, u2  5 2   7  3 d  3  (2)  5

u1  3 1  11  14, u2  3 2   11  17, d  17  14  3

u1  6  11 1  5, u2  6  11 2   16, d  16  (5)  11

u1  2a  2 1  1  2a  3, u2  2a  2 2   1  2a  5, d  2a  5  2a  3  2 © Oxford University Press 2019

u6  u1  d 6  1  u1  7 5  u1  35  37

 un  2  7  n  1  7n  5 6

u5  u1  d 5  1  0  u1  4d  0

u15  u1  d 15  1  180  u1  14d  180 Subtracting the first equation from the second: 10d  180  d  18 and substituting this into the first equation, u1  4 18   72 7 Let the three terms be a, a  d, a  2d  a   a  d    a  2d   3a  3d  24  a  d  8 and a  a  d   a  2d   640 Substituting the first equation into the second,

a  8  a  2  8  a  640  8a 16  a  640

 16a  a2  80  a2  16a  80  0   a  20   a  4   0 so a  4 or a  20 If a  4, d  12 so the numbers are -4, 8, 20 If a  20, d  12 so the numbers are 20, 8, -4

Let the three terms be a  d, a, a  d Sum of terms 3a  24  a  8

Product of terms a a2  d2  640 Substitute for a and solve

 d  144  d  12 Substituting for a and d in a  d, a, a  d the three numbers would either be 8

In year 2017, Jung Ho earned 38000  17 500  46500

38000  1.5=57000  38000  500n  57000  n  38 so in the year 2038

9 a This is an arithmetic series with u1  3, d  3  3  6

un  9  6n  93  6n  102  n  17 © Oxford University Press 2019

Using the formula Sn  S17 

17 17 3  (93)  2  90  765 2

b This is an arithmetic series with u1  31, d  40  31  9

un  9n  22  517  9n  495  n  55  S55 

55 55 31  517  2 548  15070 2

c This is an arithmetic series with u1  a  1, d  a  2  (a  1)  3

un   a  1   n  1  3  a  146  a  3n  4  a  146  3n  150  n  50  S50 

50  a  1  a  146  25 2a  145  50a  3625 2

10 a Since 3r  8 is linear relation this is an arithmetic series with 50 terms.

u1  3  8  5 u50  150  8  142 S50  b

50  5  142  3425 2

Since 7  8r is linear relation this is an arithmetic series with 100 terms.

u1  7  8  1 u100  7  800  793 S100  c

100  1  793  39700 2

Since 2ar  1 is linear relation in r , a is a constant this is an arithmetic series with 20 terms.

u1  2a  1 u20  40a  1 S20 

20 2a  1  40a  1  420a  20 2

11 a This is an arithmetic sequence with u1  4, d  5 Using the formula Sn   S15 

n 2u1  (n  1)d  2

15 2  4  5  14  465 2

b This is an arithmetic sequence with u1  3, d  8

Using the formula Sn   S10 

n 2u1  (n  1)d  2

10 2  3  9  8  390 2

c This is an arithmetic sequence with u1  1, d  5 Using the formula Sn   S20 

n 2u1  (n  1)d  2

20 2  1  5  19  930 2

12 u5  u1  4d  19 u10  u1  9d  39  u10  u5  5d  20  d  4  u1  19  4d  3  S25 

25 2  3  24  4  1275 2

u3  u1  2d  8 10 2u1  9d   230  2u1  9d  46 2 Multiplying the first equation by 9: 9u1  18d  72 S10 

Multiplying the second equation by 2: 4u1  18d  92 Subtracting: 5u1  20  u1  4

13 2  4  6  12  416 2

14 S1  6 1  3 1  3  u1  3 2

S2  6 2   3 2   12  12  0 2

So S2  S1  u2  3 d  u2  u1  3  3  6 The first four terms of the sequence are 3, -3, -9, -15

15 S  1  3  5  .  299 There are 150 odd numbers since 2n  1  299  n  150 Using the formula Sn  S150 

150 1  299  22500 2

Exercise 1C 1 a

u5  34  81 © Oxford University Press 2019

7 1 u6  63    3 27  

81 1  2  243 6 6

1 1 u7  243     192 6

a 2 1   6 a 3 4

a  1 a    2  3 162

0.02  3n 1  393.66  3n 1  19683 Using solve or Nsolve (depending on GDC type) n = 10 b

1 1 64    2 128    26  21 n  27  7  n  7  n  14 or using technology

u4  u1r 3  6 u7  u1r 6  48 

u1r 6 48  r3  8r 2 3 u1r 6

u3  u1r 2  6 u5  u1r 4  54 

u5 u1r 4 54   r2   9  r  3 u3 u1r 2 6

5 2 3  162 depending on which ratio is used  3

u1  9 u5  u1r 4  9r 4  16 16 2 2 3 r   9 3 3 So two different sequences arise depending on which common ratio is used. In either case, the seventh term is  r4 

 2 3 64 u7  u1r 6  9      3 3  

a2 a4  3a  1 a  2

  a  2    a  4  3a  1 2

 a2  4a  4  3a2  11a  4  2a2  15a  8  0  2a  1  a  8   0 1 or a  8 2 1  2 1 2 If a   , r   3 2  1 3   1  2 a

a 1 a 2  a 1 a 1

  a  1   a  2   a  1 2

 a2  2a  1  a2  3a  2  5a  1  a 

1 1 3 r  5  1 2 1 5

32  2  4  u1         3 5 135     9 a

 1 1     3   182  S6  3  81  1 1     3 b

1 1  2  8 1 1 2

1 1   7   19608  57.2  72  1 343 1 7

Or using technology b

10n  1 5  10n  1 11 u1  3 10  1 9

1 243 1 1 r6  r  729 3 Therefore there are two possible common ratios, each corresponding

u7  u1r 6  3r 6 

to a different sum to infinity 3 9   1 4 1     3 1 3 9 r  : S   1 2 3 1 3 r 

2  1 1  3  1 u2  S2  S1      1      1   2  2   4 3  1 2  3  1 u3  S3  S2      1      1   2 2 8     

1 The terms are in geometric progression with r   . To see this in general, note 2 n  1 n 1   1 n  1 n 1  1 un  Sn  Sn 1      1      1          2  2  2    2  n 1

3  1  1  1         1      2  2  2  2  i.e. the form of a general term in a geometric progression with first term 

3 1 and common ratio  2 2

u3 28 1  a  1 a u2 28

28  28  28 1  a  147 1 a 28   28a  91 1 a  28  28a 1  a  91 1  a  S3 

 28  28a  28a2  91  91a  28a2  63a  63  0  4a2  9a  9  0   4a  3   a  3   0 3 or a  3 4 1  a  1  0  a  2 for convergence

14 Let the three pieces have lengths u1, u2 and u3 © Oxford University Press 2019

u3  u1r 2  2u1  r 2  2  r  2 Since the length of the pieces must sum to 2,

u1  2u1  2u1  3  2 u1  2 2

x  x  x  x  x    1     1    1    1    1  . 2  2  2  2  2 

x  The common ratio is    1  2  Therefore the series converges when x     1  1 2   

x 1  1 2  2  x  2  2  1 

 4  x  0 When x  0.8, u1  1 and r  0.6  S 

Exercise 1D 1 a

8 220  290  2040 2

 20n  80 n4 so 2014

2 Let Jane's starting salary be S

so Jane's starting salary was €42150 to the nearest euro 3 a b

2  22  23  24  30 2  22  23  24  .  2n  106

The left hand side is a geometric series with first term 2 and common ratio 2 © Oxford University Press 2019

2(2n  1)  106 2 1  2(2n  1)  106 

Using GDC Answer: 19 generations 4

10 2  200  9  20  2900 2

so 2.9kg On the first trial she uses 100g of sugar and on the second she uses 110g. Thereafter, if the sequence is to become geometric the common ratio is 1.1 1.1n  1  1.5 1.1  1  1.1n  2.5

Using GDC n  9.614 so 9 trials In general, the geometric model is not reliable, since if Prisana were to carry out a large number of trials then the cake will become excessively sweet (since geometric growth is greater than linear growth) In fact, the ratio of sugar to flour would eventually become 1 (i.e. the mix is entirely sugar) in the (albeit unrealistic) case that Prisana carries out the trial a large number of times

Second: 12  12  2 2

 2  2       1 2    2 

1 1 1      2 2 2

3 1 1 3 7  3      3 2  7 2 2  2  1  2 2 2 2 2   4 

The length converges to a finite value since the common ratio between two consecutive side lengths that are one.

1 Area of triangle = base  height 2 Required area

 1   1   1  1 2  1  1 1 1 1                 2 2 4 8  2 2 2  4 2 8 2   2 3 4 5 6 7 1 1 1 1 1 1 1 1   1                     2  2 2 2 2 2 2  2   8   1   1 1      8  2    1  1     1      0.996 1 2 2  1   2   2

  1  1  S     1 1 2   1   2 

6 a Interest 12% pa  1% per month Let the payment per month be x . Interest is compounded monthly After one month the amount due is

1500  1.01  x After 2 months the amount due is

1500  1.01  x   1.01  x  1500 1.01  1.01 x  x 2

After 3 months the amount due is

 1.01 x  x  1.01  x  1500 1.01  1.01 x  1.01 x  x 3

After 24 months the amount due would be

  23 22  x  1.01  1.01  .  1   0   Geometric series    1.0124  1   x 0  1.01  1 

Using technology Monthly payments of $70.61 b Total amount paid

$70.61  24  $1694.64 =$1695 7 a

n 2  30  6  n  1  570 2

 60n  6n  n  1  1140  n2  9n  190  0   n  19   n  10   0  n  10

3  0.95 10  12.5  12.5m

Rapid: 200  10 0.05 200  300 so $300

 282.11975. so $282

Rapid/Quick: 100  10  0.05 100  100 1.035

Rapid: 200  25 0.05 200  450 so $450

Rapid/Quick: 100  25  0.05 100  100 1.035

The investments will be approximately equal when After n years Rapid investment: 200  10n Quick investment: 200  1.034n Rapid/Quick : 100  5n  100  1.035n Using tables on GDC: After 21 years the three investments yield approximately the same amount.

Suppose Karim invested $x in savings, therefore $  x  1000 in bonds

and $  4000  2x  in shares 75  0.015  x   0.025  x  1000  0.01  4000  2x   90  0.06 x  x  1500 so $1500 in savings, $2500 in bonds and $1000 in shares b

Now Karim is investing $1500 in savings for 10 years, $990 in savings for 9 years and $2500 in bonds for 10 years. Therefore, 1500  10  0.015 1500  990  9  0.015  990  2500 1.025

so $6048.86 =$6049 to the nearest dollar c

 2500  10 0.0152500  6048.86136.  26.3500.

x(1  0.375  0.3752  0.3753 ) , where x is the amount administered each time.

x(1  .375  .3752  .  0.37539 )  8  1  0.37540   x 8  1  0.375  8 1  0.375 x 1  0.37540

5 mg should be administered each time. c The amount of medication in the bloodsteam after n administrations is given by

 1  0.375n  5 7  1  0.375  7 1  0.375  1  0.375n  5 7 1  0.375   0.375n  1  5

Using technology to solve: There are 7mg/ml drug in the bloodstream after the third administration.

A general odd number can be written in the form 2k  1 with k 

  a  b  a2  2ab  b2  a2  2ab  b2  2a2  2b2  2 a2  b2 2

 Consider two general odd numbers 2n  1 and 2m  1, n, m  Then,

2n  1 2m  1  4nm  2n  2m  1  2 2nm  n  m  1  2p  1 p  2nm  n  m   2p  1 is an odd number 3

A four digit number represented by a3a2a1a0

not to be confused with a product 

can be written in the form

N  a3  103  a2  102  a1  10  a0 You are given that a3  a2  a1  a0  9m, m 

 N   999  1 a3   99  1 a2  9  1 a1  a0   999a3  99a2  9a1    a3  a2  a1  a0   9 111a3  11a2  a1   9m  9(111a3  11a2  a1  m) i.e. if 9 divides the sum of the digits the number itself is divisible by 9 Hence 3978, 9864 and 5670 are divisible by 9 but 5453 and 7898 are not

 a2d 2  2abcd  b2c 2  b2d 2  2abcd  a2c 2  a2d 2  b2c 2  b2d 2  a2c 2

 a2 c 2  d 2  b2 c 2  d 2

1 2 1 2 1 2       . 3 9 27 81 243 729

1 1 1 2 2 2    .     . 3 27 243 9 81 729 2 4  1 1  1  1  1 2  1 1 1 1 1 S         .  2         .  9 9 9 9 9  3 3 3 3 3   S 

Two different infinite geometric series, each with common ratio

and so both series converge.  1   1   3    S    2 9   1  1   1  1  9 9   1 9 1 9 1      2    3 8 9 8 8 6

Consider an arbitrary integer n  . Then,

 n2  n2  2n  1  n2  2n  1

1 1 1   n 1 n n 1

n  n  1   n  1  n  1  n  n  1

n  n  n2  1  n2  n

1 1 1 62  1 37 37      5 6 7 6 62  7 6 29  174

Area of trapezium:

ab ab h  a  b 2 2

Similarly, the area in terms of the triangles BAE, BEC and EDC are 1 1 1 1 ab  c 2  ab  ab  c 2 2 2 2 2 Equating the areas,

2 1 2 c   a  b   2ab  c 2 2 2  a2  2ab  b2  2ab  c 2

Suppose for the sake of contradiction that n2 is odd but n is even Then n2  2m  1 for some m 

and n  2k for some k 

But then n2  2k   4k 2  2m  1 2

4k 2 is even but 2m  1 is odd, so this is a contradiction  n2 is odd  n is also odd

Assume for the sake of contradiction that

are coprime (i.e. they have no common factors). a2  a2  3b2 b2 If p is a prime number and p divides a2 , where a   , then p must divide a. Therefore, a must be a multiple of 3  a  3k for some Then, 3 

k  . This implies 9k 2  3b2  b2  3k 2 so b is also divisible by 3. Therefore 3 is a common factor of a and b. But we assumed that a and b have no common factors, so this is a contradiction.

Suppose for the sake of contradiction that 5 2 is irrational Then

a where b are relatively coprime (i.e. share no common factors)

2 can be written in the form

 a5  2b5 so 2 divides a  a  2m for some m  5

 b  2 m so b is even which means that b is also even. So 2 divides both a and b, but it was assumed that a and b shared no common factors. This is a contradiction. 4

Suppose for the sake of contradiction that there exist p, q  such that p2  8q  11  0  p2  8q  11 so p is an odd integer  p  2k  1 for some k   2k  1  8q  11 2

 4k 2  4k  1  8q  11

 4 k 2  k  2q  10 2

but LHS is even whereas RHS is odd; this is a contradiction

Suppose for the sake of contradiction that for some a, b  , 12a2  6b2  0 2

 12a2  6b2  2a2  b2  2 

a2  a  a    2  , b2  b  b

a contradiction since we know that 6

2 is irrational.

Suppose for the sake of contradiction that for a, b, c  , the equation a2  b2  c2

You are given that a2  b2  c2, where a, b, c 

We are required to prove that either a or b must be even. Assume that both a and b are odd

a  2p  1 and b  2q  1, p, q   a2  b2  2p  1  2q  1 2

 4 p2  4 p  1  4q2  4q  1  2(2 p2  2 p  2q2  2q  1)  2n, n  You know that a2  b2  c 2 and c  2k  1, k 

c 2  2k  1  4k 2  4k  1  2 2k 2  2k  1  2m  1, m  2

The left-hand side is an even number and the right-hand side represents an odd number. This is a contradiction. Now let us assume that both a and b are even a  2p and b  2q

a2  b2  2p   2q   2 2p2  2q2  2s, s  2

 2s  2m  1 The left-hand side is an even number and the right-hand side represents an odd number which is a contradiction Hence, we have proved that precisely one of a or b must be even. 7

Suppose there exists n, k 

such that n2  2  4k

Then n must be divisible by 2 and can be written in the form n  2m with m   4m2  2  4k 1 2 But the left-hand side is an integer whereas the right-hand side is  m2  k  

not; this is a contradiction

Suppose p is irrational, q is rational and for the sake of contradiction that p  q is rational. Then, a c q and p  q  for some a, b, c, d  b d c c a bc  ad  p  q     d d b bd But by assumption, p was irrational. This is a contradiction.

and suppose for the sake of contradiction that m2  n2  1

Then, m2  n2   m  n   m  n   1 Since m, n 

The product of two positive integers can only give 1, if both are 1 or both are  1. i.e. m  n  m  n  n  n This is a contradiction since n 

Take any prime number: the number is certainly divisible by itself but is still a prime

Take n  4 : 24  1  16  1  15  35

Take the same example as in part c.

1  2  3  6, not divisible by 4

1  2  3  4  10, not divisible by 4

Exercise 1G 1 a i

b based on line divisions 1 3 5  7  9  7 5  3 1 based on colour

1  3  5  7  9  11  9  7  5  3  1

c Organizing our findings

13 1  1 4 13 5 3 1  4  9 1  3  5  7  5  3  1  9  16 1  3  5  7  9  7  5  3  1  16  25 1  3  5  7  9  11  9  7  5  3  1  25  36 . . . 2 1  3  5  .  2k  1  2k  1  k 2  (k  1)2 Conjecture: P(n): 2 1  3  5  .  2n  1  2n  1  n2  (n  1)2, n 

  d LHS = 2 1  3  5  .  2n  1  2n  1  sum of first n odd numbers 

n   2  1  (2n  1)   2n  1 2   n(2n)  2n  1  2n2  2n  1  n2  n2  2n  1  n2  (n  1)2 e P(n): 2 1  3  5  .  2n  1  2n  1  n2  (n  1)2, n 

When n  2 LHS = 2(1)  3  5 RHS= 12  22  5 © Oxford University Press 2019

LHS=RHS therefore P(1) is true. Assume that P(k) is true for some k  2, k 

i.e. 2 1  3  5  .  2k  1  2k  1  k 2  (k  1)2 Required to prove that P(k+1) is true i.e. 2 1  3  5  .  (2k  1)  (2k  1)  2k  3   k  1  (k  2)2 using the assumption 2

LHS=  2 1  3  5  .  (2k  1)  2(2k  1)  2k  3  2 1  3  5  .  (2k  1)  (2k  1)  4k  4  k 2  (k  1)2  4k  4  (k  1)2  k 2  4k  2   k  1  (k  2)2 2

Since P(2) was shown to be true, and it was shown that if P(k) is true, where k 

then P(k+1) is true, it follows by the principle of mathematical induction that P(n) is true for all n  2 a

P  n : 12  22  32  .  n2 

1 1  n  n  1  n   3 2 

When n  1, LHS  12  1 1 1 1  3 1 1  1 1 1    2     1 3 2 3  2 LHS = RHS  P(1) is true. RHS 

Assume the statement is true for n  k , where k 

Required to prove that when n  k  1, 12  22  32  .  k 2   k  1  2

1 3  k  1  k  2  k  2  3  

LHS = 12  22  32  .  k 2   k  1

2 1 1  k  k  1  k     k  1 3 2 

1   1   k  1  k  k     k  1  3 2        1 1   k  1  k  k    3  k  1  3 2    1 k  k  1  k 2  2  3k  3  3   1  2 7k    k  1  k   3 3 2   

 2k 2  7k  6  1 k  1    3 2  

  k  2  2k  3  1   k  1  3 2  

 2k  3  1 k  1  k  2      3 2  

1 3  k  1  k  2  k  2  3  

=RHS Since it was shown that P(1) is true and that P(k + 1) is true given P(k) is true for k  it follows by the principle of mathematical induction that P(n) is true for all n  b

P  n : 1  4  9  16  .   1

When n  1 LHS = 1 RHS=  1

1 2 Assume the statement P  k  is true for some k 

1  4  9  16  .   1

When n  k  1, LHS = 1  4  9  16  .   1

  k 1  k  2(k  1)    1  k  1   2   k 1  k  2    1  k  1    2 

2 i.e. P  k   P  k  1 Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers

RHS  20 1  1  2  1  1 LHS  RHS  P(1) is true

Assume that P  k  is true for some k 

When n  k  1 k 1

 2k 1  2k 1  1  2k 1  2k 1  2k 1  1  2  2k 1  1  2k 2  1

i.e. P  k   P  k  1 Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all natural numbers

P  n : 9n  1 is divisible by 8 (for n  )

P(n) : 9n  1  8 A, for n  , A  When n  0 LHS = 90  1  0 = 8  0  P(1) is true Assume P  k  to be true for some k  i.e. 8 divides 9k  1  9k  1  8m for some m  Then, 9k 1  1  9  9k  1  9  8m  1  1  9 8m  9  1  8  9m  8  8  9m  1 so 8 also divides 9k 1  1 i.e. P  k   P  k  1 Since P 0 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all natural numbers

P  n : 1  2  3  .  n  3

LHS = 13  1 12 1  1

LHS =RHS  P 1 is true Assume P  k  is true for some k  k 2  k  1

i.e. 13  23  33  .  k 3 

13  23  .  k 3   k  1  3

 k  1  k 2  4k  4 2

 4 i.e. P  k   P  k  1

Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers.

P  n : n3  n =3A, for n  , A 

When n  0: 10  1  0 = 3  0  The statement P  0 is true Assume P  k  is true for some k  k 3  k  3m for some m   k 3  3m  k When n  k  1,

LHS =  k  1   k  1  k 3  3k 2  3k  1   k  1 3

 3m  3 k 2  k  3 m  k 2  k , m  k 2  k  i.e. P  k   P  k  1

Since P 0 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all natural numbers

1 1 1 1 n ,    .   12 2  3 3 4 n  n  1 n  1

When n  1: 1 1  12 2 1 1 RHS=  11 2 LHS=RHS  P 1 is true LHS 

Assume P  k  is true for some k  i.e.

1 1 1 k   .   12 2 3 k  k  1 k  1

When n  k  1, LHS 

1 1 1 1   .   12 23 k  k  1  k  1  k  2  use assumption

k 1 1  1     k   k  1  k  1  k  2  k  1  k  2 

1  k  k  2  1  1  k 2  2k  1       k 1 k 2 k 2   k 1

2 1   k  1  k  1 k 1   k  1  k  2  k  2  k  1  1   i.e. P  k   P  k  1

Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers

P  n : n3  n = 6A for all n 

When n  1 13  1  0  0  6  P 1 is true Assume P  k  is true for some k 

k 3  k  6m for some m   k 3  k  6m When n  k  1,

  k  1  k 3  3k 2  3k  1   k  1 2

 k  6m  3k  2k  6m  3k  k  1 but k  k  1 must be an even number since any pair of consecutive natural numbers contains an even number  k  k  1  2r for some r 

  k  1   k  1  6  m  r  which is divisible by 6 3

i.e. P  k   P  k  1 Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers

P  n : 2n2  32n1 =7A n 

When n  1 LHS=212  32 1  23  33  8  27  35  7  5  P 1 is true Assume that P  k  is true for some k  2k 2  32k 1  7m for some m 

 2k 2  7m  32k 1

When n  k  1, LHS =2

2 k 1 1  3    2  2k 2  9  32k 1

 2 7m  32k 1  9  32k 1 2 k 1

 7 2m  32k 1 where 2m  32k 1  so P  k   P  k  1 Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers.

P  n : 12  32  52  .  2n  1  2

n 2n  1 2n  1 3

When n  1 LHS =12  1 RHS 

LHS =RHS  P 1 is true Assume that P  k  is true for some k 

i.e. 12  32  52  .  2k  1  2

When n  k  1 LHS =12  32  52  .  2k  1  2k  1 2

 k 2k  1  3 2k  1 

2k 2  5k  3 3 2k  1 2k  3  k  1 3

 k  1 2  k  1  1 2  k  1  1

3 i.e. P  k   P  k  1

Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers.

 r  r  1  3  n  1  n  2 r 1

 r  r  1  1 1  1  2 r 1

1 RHS = 1  1 1  2   2 3  P 1 is true

Assume P  k  to be true for some k  i.e.

 r  r  1  3  k  1  k  2 r 1

When n  k + 1, k 1

LHS = r  r  1  r 1

 r  r  1   k  1  k  2 r 1

k   k  1  k  2    k  1  k  2  3  k  1  k  2 k  3   k  1  k  1  1    3 3 i.e. P  k   P  k  1

Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers.

 r  r  1  n  1 r 1

 r  r  1  1 1  1  2 r 1

1 11  P 1 is true RHS =

Assume P  k  is true for some k  i.e.

 r  r  1  k  1 r 1

When n  k  1 LHS =

 r  r  1   r  r  1   k  1  k  2 r 1

k 1   k  1  k  1  k  2  

1  1  k   k 1 k  2

1  k  k  2  1    k  1  k 2 

1  k 2  2k  1    k 1 k 2 

2 1   k  1  k  1  k 1 k  2  k  2    P(k )  P(k  1)

Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers.

Best proved by direct argument:

 4n  3   4n  3   4n  3  4n  3  4n  3  4n  3   8n  6   48n  12(4n) so is always divisible by 12 2

(induction amongst other methods is also valid) b

False: substituting n  1 gives 75 which is not prime

Best proved by induction:

P  n  : 13  33  .  2n  1  n2 2n2  1 3

When n  1 LHS= 13  1

RHS=12 2  12  1  1 LHS=RHS  P 1 is true:

Assume the statement P  k  is true for some k 

i.e. 13  33  .  2k  1  k 2 2k 2  1 3

When n  k  1 LHS =13  33  .  2k  1  2k  1 3

 2k 4  k 2  8k 3  12k 2  6k  1  2k 4  8k 3  11k 2  6k  1

(use factor theorem to factorize or expand right hand side of P(k+1) to obtain same polynomial)

  k  1 2k  6k  5k  1 3

  k  1  k  1 2k  4k  1 2

  k  1 2  k  1  1

so P  k   P  k  1 Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers.

Best proved by induction: P  n  : 1  2  2  3  3  4.   n  1  n 

When n  1 LHS= 0  1  0 RHS=

 P 1 is true Assume the statement P  k  is true for some k  i.e. 1  2  2  3  3  4.   k  1  k 

When n  k  1 LHS= 1  2  2  3  3  4.   k  1  k  k  k  1

 k  1 3 k  k  1 (k  1)  3k  k  1 3 (k  1)(k  k  1  3k )

3 so P  k   P  k  1 Since P 1 is true and P  k   P  k  1 for k 

then by the principle

of mathematical induction, the statement is true for all positive integers.

Best proved by direct argument: © Oxford University Press 2019

n3  n  n n2  1   n  1 n  n  1 this is the product of three consecutive positive integers (in the case n  1, 0 is divisible by 3 so done) Three consecutive positive integers always include a multiple of 3, so the product is always divisible by 3

n! n  1  1  n  n!

 n  1 ! n  1   n  1  n  1 !

 n  1 ! n  1   n  1  n  1!

n! n  1  1   n  2 n!

8! 8  7  6!   14 4  6! 4  6!

4!  5!  4 3! 5! 4 2    3!  6! 3! 6  5! 6 3

10! 8!  10! 8 7 6! 11! 6! 11 10! 6!

 n  1 !   n  1 !  n  1 ! n!1   n  1 

n ! 1  n  1 n!  1  n !  1 n!  1

2n  2 ! n!  (2n  2)(2n  1)  2(2n  1) 2 (n  1)2 n 1  n  1 ! 2n !

 n2  n  132  0   n  12   n  11  0 n  0 so n  12

6 16  n  1 !  5n!  n  1 !

 16  5n   n  1 n  n2  6n  16  0   n  8  n  2  0 n2 7 a

13! 4! 4! 3! 2! 4!  165888

26  25  24  10  9  1404000 23

b Number of ways of choosing all boys = 13 C5 Number of ways of choosing all girls

Number of ways of choosing at least one boy and at least one girl =

C5 10 C5  32110

6  6  5  4  720

Last digit must be 0, 4 or 8 6  7  7  3  882

Last digit must be 0

6  7  7  1  290

11 6 C4  15 12 There are 5C3 ways to choose the drivers. Then, there are 9 ways to choose passenger for small car. This leaves 8 persons to choose 4 passenghers for second car and the rest go in the third car. 

C3  3!  9 8 C4 4 C4  37800

11 10 9   x   .  x  11 10  x   1  11          2! 3!  3  3  3 2

11x 55x 2 55x 3    . 3 9 9

7 6  5  x   . x   x  7   6   x  1    1  7         2 2!  2  3!  2 2 2 3 7x 21x 35x 1    . 2 4 8 7

2 2   8   x    x 1  2  x x   

2  8 7 6   2 3  .   2   8  7   2   x 8 1  8  2       2 2!  x 2  3! x  x    8 6 4 2  x  16 x  112x  448x  .

9  4  C2  a  2   880a5 a  3

5  2y  2 3 C3  x      448x y  x 

General term is given by r

 2  0   2   Nx x   Comparing powers of x 12  r  2r  0 12

8  2  12 C8  x    2   7920  x 

x x     2    16 1   5 10    

2 3 4   x  4  x  4  x  4  x     16  4 C0  4 C1    C2    C3    C4         10   10   10   10     2 x 3x 2 x3 x4   16 1      5 50 250 10000   32x 24x 2 8x 3 x4  16     5 25 125 625 4 0.05    1.99    2   5  

5 25  15.68239 to 5d.p.

General term is given by r

 1 6     Nx  x Comparing powers of x 12  2r  r  6 6

 1 6     15x  x 5

 x5  5x 3y  10xy 2 

y  y y 5 4 y  3 y  2 y   x    x  5x    10x    10x    5x      x x x x x           x 10y 3 5y 4 y 5  3  5 x x x

2x  y   x5  5x 3y  10xy 2   Term in x 3y 2 is

10y 3 5y 4 y 5   3  5 x x x 

y 5x 3y  5x 3y 2 so 5

8  n! 4  n!  n  3 !3! 3     n  3 !

 3    2   10  3    2   5  3   2     2  5

 9 3  45 2  60 3  60 2  20 3  4 2  89 3  109 2 4

  1  5   2    2  5 5     

 5  5 2    5   5     

8 10 12 4 10 1    5 5 25 25 161 44   10 25 25  4

 5    5   5  525 5  125 5  3

C0  2 n C1  4 n C2  8 n C3  .   1 2r n Cr  .   1 2n n Cn r

C0 n C1 n C3  .   1 n Cr  .   1 n Cn  1  1  0 r

Exercise 1J 1 a

 1  2 x 2   1  2  3 x 3  . 1 1  1  x   1   1 x      1 x 2! 3!  1  x  x2  x3  .

2!  1  4x  12x 2  32x 3  . c

Using the answer to part a and substituting 2x for x, 1 2  2 1  2x   2 1  2x  4x 2  8x 3  . 1  2x  2  4x  8x 2  16 x 3  .

  3  4  x 2   3  4  5  x 3  .  2 1   3   x        2! 3!   2 3  2  6 x  12x  20x  .

1  2x  1  2x  2 1

1 1 1 1 3 2 2 2 2 2 2 1       2x 3  .  1  2 x   2x         2 2 3! 1 1  1  x  x 2  x 3  . 2 2

31 31 1 22 22 2 3      x 3  . 2 1 x  x     2 2! 3!

3 x 3x 2 x 3    . 2 8 16

 1 3  2 2  1   3x 2   1      3x      2 2   3x 27x 2 135 3 1   x  . 2 8 16 d

 1 3 5  2 2 2     3x 3  .   3!

  1 2 1 2 5   3 3 3 3 3 1  x2       x 3  .   2 1  x       3 2 3!     1 1 2 5 3    2 1  x  x  x  .  3 9 81   2 x 2 x 2 10 x 3 2    . 3 9 81 1 1  1 x  1  x 2 1  x  2 1 x

  1 1 1 1 3   2 2 2 2 2 1  x2       x 3  .. 1  1 x   1  x       2 2 3! 2        x x2 x3 x 3x 2 5x 3  1     .  1     .  2 8 16 2 8 16   

  1 3 5   2 2 2     x 3  ..  3!  

 1 3  2 2    x2  2

  2  3 x 2   2  3  4 x 3  .   x 1   2 x   2! 3!    x 1  2x  3x 2  4x 3  .   x  2x 2  3x 3  4x 4  . 5

2  3  4  5   3x 3  .  1  3x   3  4   3x  1   3            8  2! 3!  2   2   2  

 1 9 x 27x 2 135x 3    .  1  8 2 2 4 

1 9x 27x 2 135x 3     . 8 16 16 32

1  4x  1  4x  2 1

1 1 2 2 1  4x 2   1   4x        2 2!  1  2x  2x 2  4x 3  .

1 1 3 2 2 2     4x 3  .   3!

4 2 6  1   96  1  4 6       5  100   100  10

5  1  1  4  2  100  2

5  1   1   1  (1  2    2   4   . ) 2 100 100      100 

 1 3  1 3 5  2 2     2  1   2x   2   2   2  2x 3 .  1      2x        2! 3!  2 3x 2 5x 3 1 x    . 2 2 b

  3x 2 5x 3  (2  3x)3 1  x    .  2 2 1  2x  

 2 2 3  3x 2 5x 3  3  2   3 x   3 2  3 x   3 x  1  x    .  2 2    8  8 x  12x 2  20x 3  . 3

+36x  36 x 2  54 x 3  . +54x 2  54 x 3  .  27 x 3  .  8  44x  102x 2  155x 3  .

Chapter review 1

S3  u1 1  r  r 2  91 9 1  r  r 2  91 r  9  9r  9r 2  91r

 9r 2  82r  9  0   9r  1  r  9   0 1 or r  9 9 Therefore there are two geometric sequences:

1 1  u4  9 9 r  9 :  u4  243 r 

u1  1 1  2  3  4  5  7  8  9  11  13  15  16  17  .  64  1  3  5  7  .  63  2  4  8  16  .  64  arithmetic series sum of first 32 odd numbers

Finite geometric series,u=2, r=2, n=6

32 2(26  1) 1  63   2 2 1  1024  126 

b  a  d, c  a  2d  a  d  12  a  12  d c a a  2d a    b c ad a  2d  (a  2d )2  a(a  d ) Substituting for a  12  d  2d   12 12  d  2

 12  d   144  12d 2

 144  24d  d 2  144  12d  0  d 2  36d  d  d  36   0 d  0, d  36 a  48 b  48  36  12 c  48  72  24

1 1 1   2  1  x 3  2x  3 1  x  3  

x 2 x 2  3  2 x  3x  2 x 2 2 x 2  5x  3

1 x 2 1 2   1  x   1  x  2 2 x  5x  3 3 3 

2 7 23 2 73 3  x x  x  . 3 9 27 81 n! 1  n  n  n  1  n 2  n  2 !2!

1 1 1  n  1 ! n  n  1  2   n  n  1  2 2 2  n  1 !

1 2 4 2 8 3  x  .  1  x  x  3 3 9 27 

 n  2 ! n!  2!  n  2 !  n  k  !  k  2 !

n! n! 1   2!  n  k  !  k  2  !  n  k  ! 2!  k  2  ! n!k !

n C0 n C1x n C2 x2  . n Cr x r  . n Cn x n

n C0  n C1  3  n C2  32  .  n Cr  3r  .  n Cn  3n

Suppose there exist integers a and b such that 14a  7b  1. Then, 2a  b 

But the left-hand side is an integer whereas the right-hand side is not. This is a contradiction. Therefore there are no such integers. 8

Suppose x  3 and 5x  7  13. Then, x 

13  7  4. But x  3, so this is a contradiction 5

Take, for example, a  0 and b  1

Take, for example, n  5 : 35  2  245  5  49 which is not prime

Take, for example, n  1:

Take, for example, n  1 : 21  1  1 and 1is not prime

2 1  1  1  1 which is rational

1  1!  22  2!  33  3!  .   nn  n!   n!

When n  1 LHS= 1  1!  1 RHS= 1!

Assume the statement P  k  is true for some k 

i.e. 1  1!  22  2!  . k k  k !   k ! When n  k  1

LHS= 1  1!  22  2!  . k k  k ! 

  k  1 k !  k  1 !    k  1 !  k  1 !    k  1 ! k 1

so P  k   P  k  1 Therefore, it has been shown that P 1 is true and that if P  k  is true then so is P  k  1 . Therefore, the statement is true

for all positive integers by the principle of mathematical induction 11 P  n : n3  2n = 3A, A 

When n  1 13  2 1  3 The statement P 1 is true Assume that P  k  is true for some k  k 3  2k  3m for some m 

 k  3m  2k When n  k  1 LHS=  k  1  2  k  1 3

 k 3  3k 2  3k  1  2k  2  3m  2k  3k 2  5k  3

 P  k   P  k  1 Therefore, it has been shown that P 1 is true and that if P  k  is true 

then so is P  k  1 . Therefore, the statement is true

for all positive integers by the principle of mathematical induction.

Assume that P  k  is true for some k  i.e.

When n  k  1, k 1

2 so P  k   P  k  1 Therefore, it has been shown that P 1 is true and that if P  k  is true then so is P  k  1 . Therefore, the statement is true

for all positive integers by the principle of mathematical induction b

n  n  1 2n  1 6

Assume that P  k  is true for some k  i.e.

k  k  1 2k  1 6

When n  k  1, k 1

k  k  1 2k  1 6

2k  1  6  k  1  6  k 1  2k  3  k  2 6  k  1  k  2 2  k  1  1  6 so P  k   P  k  1 

k 1 2k 2  7k  6 6

Therefore, it has been shown that P 1 is true and that if P  k  is true then so is P  k  1 . Therefore, the statement is true

for all positive integers by the principle of mathematical induction

Assume that P  k  is true for some k  i.e.

When n  k  1, k 1

 k  1  k 2  4k  4 2

4 so P  k   P  k  1 Therefore, it has been shown that P 1 is true and that if P  k  is true for some k 

then so is P  k  1 . Therefore, the statement is true

for all positive integers by the principle of mathematical induction n

  r  r  1  r  2  r 1

n  n  1 2n  1 2

n  n  1  2 2n  1  4 4 n  n  1 2 n  5n  6   4 

n  n  1  n  2   n  3 4

13 a ‘harmonics’ consists of 9 different letters, so there are 9! arrangements. b 5 digit numbers: 4 ways of choosing first digit (bigger than 3) Each of the next three digits can be chosen in 7 ways The last digit can be 0 or 5 These numbers include 30000 which is not wanted In all there are 4  73  2  1 five digit numbers 6 digit numbers 6 ways of choosing first digit 7 ways of choosing each of the next four digits 2 ways of choosing last digit Divisible by 5  final digit is 0 or 5 In all there are 4  74  2 six digit numbers 7 digit numbers 6 ways of choosing first digit © Oxford University Press 2019

7 ways of choosing each of the next five digits 2 ways of choosing last digit In all there are 4  75  2 six digit numbers

Answer = 4  73  2  1  4  74  2  4  75  2

 1371  19208  134456  155035

c The only possibilities would be to have 3 women and 2 men or 4 women and 1 man 4

C3 7 C2 4 C4 7 C1  4  21  1  7  91

a2  b2   a  b  a  b  2x  2y   4xy

a3   x  y   x3  3x2y  3xy 2  y 3 3

b3   x  y   x3  3x2y  3xy 2  y 3 3

 a3  b3  x3  3x2y  3xy 2  y 3  x3  3x2y  3xy 2  y 3

  a  b  3x  y 2 2

But, a2  ab  b2  3x 2  y 2 So,

a3  b3   a  b  a2  ab  b2

a4   x  y   x 4  4x3y  6x2y 2  4xy 3  y 4 4

b4   x  y   x 4  4x 3y  6 x 2y 2  4xy 3  y 4 4

 a4  b4  x 4  4x 3y  6 x 2y 2  4xy 3  y 4  x 4  4x 3y  6 x 2y 2  4xy 3  y 4 3

 2y  4 x 3  4 xy 2

  a  b  a3  a2b  ab2  b3

Conjecture: an  bn   a  b an1  an2b  an3b2  .  abn2  bn1

P  n : an  bn   a  b an1  an2b  an3b2  .  abn2  bn1

When n=2 LHS=a2  b2 RHS   a  b   a  b   a2  ab  ab  b2  LHS P(2) is true Assume that P  k  is true for some k 

i.e. ak  bk   a  b  ak 1  ak 2b  ak 3b2  .  abk 1

ak 1  bk 1  a ak  bk 1

 ak 2b  ak 3b2  .  abk 1  bk   bk 1 

 a ba  a  a  b  a  a b  a   a  b a  a  a b  a    a  b  a  a b  a b  a  a  b a

  ab  b b  .  ab   b  a  b  b  .  ab   b    .  ab  b 

so P  k   P  k  1 Therefore, it has been shown that P2 is true and that if P  k  is true for some k 

, k  2 then P  k  1 is also true. Therefore, the statement is true

for all positive integers greater than 2, by the principle of mathematical induction

15 The difference between the coefficients must be the same n Cr n Cr 1 n Cr 1 n Cr n! n! n! n!    r !  n  r  !  r  1 !  n  r  1 !  r  1 !  n  r  1 ! r !  n  r  !

 r  1  n  r  1   r  1  r    n  r  1  n  r    r  1  n  r  1  2  r  1  n  r  1  r  r  1   n  r  1  n  r   0   n  r  1 3r  2  n   r 2  r  0 which after expanding and simplyfing gives n2  4r 2  2  n  4r  1  0

A B C   1  2x 1  x 1  x

 2  x  7 x 2  A 1  x  1  x   B 1  2 x  1  x   C 1  2x  1  x  Set x  1 :  4  2C  C  2 Set x  1 :  6  6B  B  1 Compare constants : 2  A  B  C  A  2  B  C  1 2  x  7x 2

1 1 2   1  2x 1  x 1  x

1  2x  1  x  1 1 1  1  2 x   1  x   2 1  x  2

 1  2x  4x 2  8x 3  .  1  x  x 2  x 3  .  2 1  x  x 2  x 3  . 2

17 Require ( 3  coefficient of term in x 5 )  1  coefficient of term in x 4 8 8 5 4 3    43  2x   1    44  2x  5  4

 3   114688  1  286720

 57344  n 2 18   1n 2 3x   495x 2 2

n  n  1  110 n2  n  110  0

 n  11  n  10  0

So n  11 or n  10

19 First part is geometric sum, a  1 , r  1.6 , n  16 Second part is arithmetic sum, a  0 , d  12 , n  16 Third part is 16  1  16

(1 mark) (1 mark) (1 mark)

Geometric sum: S16 

1.616  1  3072.791 1.6  1

Arithmetic sum: S16 

16 2  0  15   12  1440 2