P1 (Cambridge)

Worksheet On

Arithmetic Series

geometric Series &

Modelling with series

Engr. Md. Nadim (ME; BUET)

Senior Teacher

A Level Mathematics (Edexcel and Cambridge)

MASTERMIND School

Cell: 01713-049960

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 1 | 9

1. For each of the following arithmetic series, write down the common difference and find the value of the 40th term. a b c

2. For each of the following arithmetic series, find an expression for the nth term in the form

.

a b

c

3. Find the sum of the first 30 terms of each of the following arithmetic series.

a b c

4. Given the first term, a, the last term, l, and the number of terms, n, find the sum of each of the

following arithmetic series. a , , b , , c , ,

5. Give the first term, a, the common difference, d, and the last term, l, find the sum of each of the following arithmetic series. a , , b , , c , ,

6. Give the first term, a, the common difference, d, and the last term, l, find the sum of each of the

following arithmetic series. a , , b , , c , ,

7. The first and third terms of an arithmetic series are 21 and 27 respectively.

a Find the common difference of the series b Find the sum of the first 40 terms of the series.

8. The nth term of an arithmetic series is given by

Find the first term of the series and the sum of the first 35 terms of the series.

9. The second and fifth terms of an arithmetic series are 13 and 46 respectively.

a Write down the two equations relating the first term, a, the common difference, d, of the series b Find the value of a and d. c Find the 40th term of the series.

10. The third and eighth terms of an arithmetic series are 72 and 37 respectively. a Find the first term and common difference of the series. b Find the sum of the first 25 terms of the series.

11. The fifth term of an arithmetic series is 23 and the sum of the first 10 terms of the series is

240. a Find the first term and common difference of the series. b Find the sum of the first 60 terms of the series.

12. Find the sum of a all even numbers between 2 and 160 inclusive. b all positive integers less than 200 that are divisible by 3 c all integers divisible by 6 between 30 and 300 inclusive.

13. An arithmetic series has common difference -11 and tenth term 101.

a Find the first term of the series. b Find the sum of the first 30 terms of the series.

Arithmetic Series

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 2 | 9

14. The first and fifth terms of an arithmetic series are 17 and 27 respectively. a Find the common difference of the series.

Given that the rth term of the series is 132, b find the value of r, c find the sum of the first r terms of the series.

15. The sum of the first six terms of an arithmetic series is 213 and the sum of the first ten terms

of the series is 295. a Find the first term and the common difference of the series. b Find the number of positive terms in the series. c Hence find the maximum value of , the sum of the first n terms of the series.

16. The sum, , of the first n terms of an arithmetic series is given by . a Evaluate . b Find the eighth term of the series. c Find an expression for the nth term of the series.

17. The first three terms of an arithmetic series are , and – respectively. a Find the value4 of the constant k.

b Find the sum of the first 25 terms of the series.

18. The fifth, sixth and seventh terms of an arithmetic series are – , and – respectively. a Find the value of the constant t.

b Find the sum of the first 18 terms of the series. 19. The third term of an arithmetic series is -10 and the sum of the first eight terms of the series is

16. a Find the first term and the common difference of the series. b Find the smallest value of n for which the nth term of the series is greater than 300.

20. The third and seventh terms of an arithmetic series are

and

respectively.

a Find the first term and the common difference of the series. b Show that the sum of the first n terms of the series is given by where k is an exact fraction to be found.

21. An arithmetic series has first term a and common difference d.

Given that the sum of the first nine terms of the series is 126, a show that . Given also that the sum of the first 15 terms of the series is 277.5, b find the values of a and d, c find the sum of the first 32 terms of the series.

22. Three consecutive terms of an arithmetic series are – , and respectively. a Find the value of the constant k. b Find the smallest positive term in the series. Given also that the series has r positive terms,

c Show that the sum of the positive terms of the series is given by – . 23. a State the formula for the sum, , of the first n terms of an arithmetic series with first term a

and common difference d. b Prove that for any arithmetic series, . c An arithmetic series has first term 40 and common difference -3. Find the sum of the positive terms of the series.

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 3 | 9

24. The first and fourth terms of an arithmetic series are x and respectively.

a Find expressions in terms of x for i the seventh term of the series, ii the common difference of the series, iii the sum of the first ten terms of the series. Given also that the 20th term of the series is 52, b find the value of x.

25. An arithmetic series has first term a and common difference d.

Given that the sum of the first twenty terms of the series is equal to the sum of the next ten terms of the series, show that the ratio .

26. The sum, , of the first n terms of a series is given by

. a Show that the sixth term of the series is 10. b Find an expression for the nth term of the series in the form . c Hence, prove that the series is arithmetic.

1. For each of the following geometric series, write down the common ratio and find the value of

the eighth term. a . . . b . . . c . . .

2. For each of the following geometric series, find an expression for the th term. a . . . b . . . c

3. Find the sum of the first terms of each of the following geometric series.

a . . . b . . . c

. . .

4. Given that the first term, , the common ratio, , and the number of terms, find the sum of

each of the following geometric series. Give your answers to decimal places where appropriate.

a b

c

d e

f

5. The second and third terms of a geometric series are and respectively.

a Find the common ratio of the series. b Find the first term of the series. c Find the sum of the first eight terms of the series.

6. The first and fourth terms of a geometric series are and respectively.

a Find the common ratio of the series. b Find the ninth term of the series. 7. The third and fourth terms of a geometric series are and respectively.

a Find the common ratio of the series. b Find the first term of the series. c Find, to decimal places, the sum of the first terms of the series.

8. The first and third terms of a geometric series are and respectively. a Find the two possible values of the common ratio of the series.

Geometric Series

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 4 | 9

Given also that the common ratio of the series is positive, b find the sum of the first terms of the series.

9. The first and fourth terms of a geometric series are and respectively.

a Find the common ratio of the series. b Find the tenth term of the series. 10. The second and fifth terms of a geometric series are and respectively.

a Find the first term and common ratio of the series. b Find the number of terms of the series that are smaller than

11. The sum of the first four terms of a geometric series is and its common ratio is

a Find the first term of the series. b Find the eighth term of the series. c Find the least value of for which the sum of the first terms of the series is greater than

12. All the terms of a geometric series are positive. The sum of the first and second terms of the series is and the sum of the third and fourth terms of the series is a Find the first term and common ratio of the series. b Find the sum of the first terms of the series.

13. For each of the following geometric series, either find its sum to infinity or explain why this cannot be found. a . . . b . . . c

d . . e

. . f

14. Find the sum to infinity of the geometric series with th term

a b

c

d

15. A geometric series has first term and common ratio

a Find the sum to infinity of the series. b Find the difference between the sum to infinity of the series and the sum of the first six

terms of the series.

16. The common ratio of a geometric series is and the sum to infinity of the series is

a Find the first term of the series. b Find the smallest value of for which the th term of the series is less than

17. The sum, of the first terms of a geometric series is given by

a Find the first term and fifth term of the series. b Find an expression for the th term of the series.

18. The first three terms of a geometric series are and respectively.

a Find the value of the constant b Find the sum to infinity of the series.

19. The third and fourth terms of a geometric series are and

respectively.

a Find the first term of the series. b Find the sum to infinity of the series.

20. The first three terms of a geometric series are and respectively,

where is a positive constant.

a Find the value of

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 5 | 9

b Find the sixth term of the series. c Show that the sum of the first ten terms of the series is to decimal place.

21. The second and fifth term of a geometric series are and respectively.

a Show that the first term of the series is b Find the value of the tenth term of the series to decimal place. c Find the sum of the first terms of the series to decimal place.

22. a Prove that the sum, of the first terms of a geometric series with first term and common

ratio is given by

.

b A geometric series has first term and common ratio √

Given that the sum of the first terms of the series is (√ ) find the value of

23. The first term of a geometric series is and the sum to infinity of the series is

a Find the common ratio of the series. b Find the third term of the series.

c Find the exact difference between the sum of the first eight terms of the series and the sum to infinity of the series.

24. The sum of the first terms of a geometric series is given by

a Show that the third term of the series is b Find an expression for the th term of the series in the form where is an exact fraction.

Answers:

1 a)

b)

c)

2 a)

b)

c)

3 a)

b)

c)

4 a)

b)

c)

5 a)

b)

c)

6 a)

b)

c)

7 a)

b)

8 first team

9 a)

b)

c)

10 a)

b)

11 a)

b)

12 a)

b)

c)

13 a)

b)

14 a)

b)

c)

15 a)

b) positive terms

c)

16 a)

b)

c)

17 a)

b)

18 a)

b)

19 a)

b) smallest

20 a)

b) show

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 1 | 9

21 a) show

b)

c)

22 a)

b) smallest +ve term

c) show

23 a)

b) Proof

c)

24 a) i)

ii)

iii)

b)

or

25

26 a) show

b)

c) Proof

Geometric

1 a)

b)

c)

2 a)

b)

c) (

)

3 a)

b)

c)

4 a)

b)

c)

d)

e)

f)

5 a)

b)

c)

6 a) b)

7 a)

b)

c)

8 a) b)

9 a)

b)

10 a)

b) terms

11 a) b)

12 a) b)

13 a) b)

c) no value d)

e) no value f)

14 a) b)

c)

d)

15 a) b)

16 a) b)

17 a) b)

18 a) b)

19 a) b)

20 a) b)

21 b) c)

22 b)

23 a) b)

c)

24 b)

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 1 | 9

Modelling with Series

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 2 | 9

Worksheet Arithmetic Series, Geometric Series & Modelling with Series

P a g e 3 | 9

Answers: