Solution of Chapter 9. Arithmetic Progressions (RD Sharma - Mathematics Book)

Exercise 9.1

1

Write the first five terms of each of the following sequences whose nth terms are :

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

2

Find the indicated terms in each of the following sequences whose nth terms are:

(i)

(ii)

(iii)

(iv)

(v)

3

Find the next five terms of each of the following sequences given by:

(i)

(ii)

(iii)

(iv)

Exercise 9.2

1

For the following arithmetic progressions write the first term a and the common difference d :

(i)

(ii)

(iii)

(iv)

2

Write the arithmetic progression when first term a and common difference of d are as follows:

(i)

(ii)

(iii)

3

In which of the following situation, the sequence of numbers formed will form an A.P.?

(i) The cost of digging a well for the first metre is Rs. 150 and rises by Rs.20 for each succeeding metre.

(ii)The mount of air present in the cylinder when a vacuum pump removes each time of their remaining in the cylinder.

4

Show that the sequence defined byis an A.P., find its common difference.

5

Show that the sequence defined by is not A.P.

6

The general term of a sequence is given by .Is the sequence an A.P..? If so, find its 15th term and the common difference.

7

Find the common difference and write the next four terms of each of the following arithmetic progressions:

(i)

(ii)

(iii)

(iv)

8

Prove that no matter what is the real number a and b are, the sequence with nth term a+nb is always an A.P. What is the common difference?

9

Write the sequence with nth term:

(i)

(ii)

(iii)

(iv)

Show that all of the above sequences form A.P.

10

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

(ix)

(x)

(xi)

(xii)

11

Find the common difference of the A.P. and write the next two terms:

(i)

(ii)

(iii)

(iv)

(v)

12

The nth term of an A.P. is 6n+2. Find the common difference.

Exercise 9.3

1

Find:

(i) 10th term of the A.P.

(ii) 18th term of the A.P.

(iii) nth term of the A.P.

(iv) 10th term of the A.P.

(v) 8th term of the A.P.

(vi) 11th term of the A.P.

(vii) 9th term of the A.P.

2

(i) Which term of the A.P. is 248?

2

Which term of the A.P.is 0?

2

Which term of the A.P. is 254?

2

Which term of the A.P. is 420?

2

Which term of the A.P. is its first negative term?

3

Is 68 a term of the A.P. ?

3

Is 302 a term of the A.P. ?

3

Is -150 a term of the A.P. ?

4

How many terms are there in the A.P.?

(i)

(ii)

(iii)

(iv)

5

The first term of an A.P. is 5, the common difference is 3 and the last term is 80; find the number of terms.

6

The 6th and 17th terms of an A.P. are 19 and 41 respectively, find the 40th term.

7

If 9th term of an A.P. is zero, prove that its 29th term is double the 19th term.

8

If 10 times the 10th term of an A.P. is equal to 15 times the 15th term, show that 25th term of the A.P. is zero.

9

The 10th and 18th terms of an A.P. are 41 and 73 respectively. Find 26th term.

10

In a certain A.P. the 24th term is twice the 10th term. Prove that the 72nd term is twice the 34th term.

11

If (m+1)th term of an A.P. is twice the (n+1)th term, prove that (3m+1)th term is twice (m+n+1)th term.

12

If the nth term of the A.P.is same as the nth term of the A.P.find n.

13

Find the 12th term from the end of the following arithmetic progressions:

(i)

(ii)

(iii)

14

The 4th term of an A.P. is three times the first and the 7th term exceeds twice the third term by 1. Find the first term and the common difference.

15

Find the second term and nth term of an A.P. whose 6th term is 12 and the 8th term is 22.

16

How many numbers of two digit are divisible by 3?

17

An A.P. consists of 60 terms. If the first and the last terms be 7 and 125 respectively, find 32nd term.

18

The sum of 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 34. Find the first term and the common difference of the A.P.?

19

The first term of an A.P. is 5 and its 100th term is - 292. Find the 50th term of this A.P.

20

Findfor the A.P.

(i)

(ii)

21

Write the expressionfor the A.P.

Hence, find the common difference of the A.P. for which

(i)11th terms is 5 and 13th term is 79.

(ii)

(iii) 20th term is 10 more than the 18th term.

22

Find n if the given value of x is the nth term of the given A.P.

(i)

(ii)

(iii)

(iv)

23

If an A.P. consists of n terms with first term a and nth term ls how that the sum of the mth term from the beginning and the mth term from the end is (a+l).

24

Find the arithmetic progression whose third term is 16 and seventh term exceeds its fifth term by 12.

25

The 7th term of an A.P. is 32 and its 13th term is 62. Find the A.P.

26

Which term of the A.P.will be 84 more than its 13th term?

27

Two arithmetic progressions have the same common difference. The difference between their 100th terms is 100, what is the difference between their 1000th terms?

28

For what value of n, the nth terms of the arithmetic progressions andare equal?

29

How many multiples of 4 lie between 10 and 250?

30

How many three digit numbers are divisible by 7?

31

Which term of the arithmetic progression will be72 more than its 41th term?

32

Find the term of the arithmetic progression which is 39 more than its 36th term.

33

Find the 8th term from the end of the A.P.

34

Find the 10th term from the end of the A.P. .

35

The sum of 4th and 8th terms of an A.P. is 24 and the sum of 6th and 10th terms is 44. Find the A.P.

36

Which term of the A.P. will be 120 more than its 21st term?

37

The 17th term of an A.P. is 5 more than twice its 8th term. If the 11th term of the A.P. is 43, find the nth term.

38

Find the number of all three digit natural numbers which are divisible by 9.

39

The 19th term of an A.P. is equal to three times its sixth term. If its 9th term is 19, find the A.P.

40

The 9th term of an A.P. is equal to 6 times its second term. If its 5th term is 22, find the A.P.

41

The 24th term of an A.P. is twice its 10th term. Show that its 72nd term is 4 times its 15th term.

42

Find the number of natural numbers between 101 and 999 which are divisible by both 2 and 5.

43

If the seventh term of an A.P. is 1/9 and its ninth term is1/7, find its (63)rd term.

44

The sum of 5th and 9th terms of an AP is 30. If its 25th term is three times its 8th term, find the AP.

45

Find where 0 (zero) is a term of the A.P.

46

Find the middle term of the A.P..

47

If the 5th term of an A.P. is 31 and 25th term is 140 more than the 5th term, find the A.P.

Exercise 9.4

1

The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds these term by 6, find three terms.

2

Three numbers are in A.P. If the sum of these numbers be 27 and the product 648, find the numbers.

3

Find the four numbers in A.P. whose sum is 50 and in which the greatest number is 4 times the least.

4

The angles of a quadrilateral are in A.P. whose common difference is 10°. Find the angles.

5

The sum of three numbers in A.P. is 12 and the sum of their cube is 288. Find the numbers.

6

Find the value of x for which (8x+4), (6x-2), (2x+7) are in A.P.

7

If x+1, 3x and 4x+2 are in A.P, find the value of x.

8

Show that (a–b)2, (a2+b2) and (a + b)2 are in A.P.

Exercise 9.5

1

Find the sum of the following arithmetic progressions:

(i)to 10 terms

(ii)to 12 terms

(iii)to 25 terms

(iv)to 12 terms

(v)to 22 terms

(vi)to n terms

(vii)to n terms

(viii)to 36 terms

2

Find the sum on n term of the A.P.

3

Find the sum of n terms of an A.P. whose nth term is given by

4

If the n sum of a certain number of terms starting from first term of an A.P. is is 116. Find the last term.

5

How many terms of the sequence should be taken so that their sum is zero?

5

How many terms are there in the A.P. whose first and fifth terms are 14 and 2 respectively and the sum of the terms is 40?

5

How many terms of the A.P.must be taken so that their sum is 636?

5

How many terms of the A.P. must be taken so that their sum is 693?

6

The first and the last terms of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

7

The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.

8

The first term of an A.P. is 2 and the last term is 50. The sum of all these terms is 442. Find the common difference.

9

If 12th term of an A.P. is 13 and the sum of the first four terms is 24, what is the sum of first 10 terms?

10

Find the sum of first 22 terms of an A.P. in which d = 22 and a22 = 149.

11

Find the sum of all natural numbers between a and 100, which are divisible by 3.

12

Find the sum of first n odd natural numbers.

13

Find the sum of all odd numbers between

(i) 0 and 50

(ii) 100 and 200

14

Show that the sum of all odd integers between 1 and 1000 which are divisible by 3 is 83667.

15

Find the sum of all integers between 84 and 719, which are multiples of 5.

16

Find the sum of all integers between 50 and 500, which are multiples of 7.

17

Find the sum of all even integers between 101 and 999.

18

Find the sum of all integers between 100 and 550, which are multiples of 9.

19

In an A.P. if the first term is 22, the common difference is – 4 and the sum to n terms is 64, find n.

20

In an A.P. If the 5th and 12th terms are 30 and 65 respectively, what is the sum of first 20 terms?

21

Find the sum of the first

(i) 11 terms of the A.P :

(ii) 13 terms of the A.P :

(iii) 51 terms of the A.P. whose second term is 2 and fourth term is 8.

22

Find the sum of

(i) The first 15 multiples of 8

(ii) The first 40 positive integers divisible by (a) 3 (b) 5 (c) 6.

(iii) All 3 – digit natural numbers which are divisible by 13.

(iv) All 3 – digit natural numbers, which are multiples of 11.

23

Find the sum :

(i)

(ii)

(iii)

(iv)

(v)

(vi)

(vii)

(viii)

24

Find the sum of the first 15 terms of each of the following sequences having nth term as

(i)

(ii)

(iii)

(iv)

25

Find the sum of first 20 terms of these sequence whose nth term is .

26

Find the sum of first 25 terms of an A.P. whose nth term is given by .

27

Find the sum of the first 25 terms of an A.P. whose nth term is given by .

28

Find the sum of the first 25 terms of an A.P. whose second and third terms are 14 and 18 respectively.

29

If the sum of 7 terms of an A.P. is 49 and that of 17 term is 289, find the sum of n terms.

30

The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

31

In an A.P., the sum of first n terms is . Find its 25th term.

32

Let there be an A.P. with first term ‘a’, common difference, d. Ifdenotes its nth term andthe sum of first n terms, find.

(i)

(ii)

(iii)

(iv)

(v)

(vi)

33

Aman saved Rs. 16500 in ten years. In each year after the first he saved Rs.100 more than he did in the preceding year. How much did he save in the first year?

34

Aman saved Rs.32 during the first year, Rs.36 in the second year and in this way he increases his savings by Rs.4 every year. Find in what time his saving will be Rs.200.

35

Aman arrange stop ay off a debt of Rs.3600 by 40 annual installments which for man arithmetic series. When 30 of the installments are paid, he dies leaving one - third of the debt unpaid, find the value of the first installment.

36

There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all the trees separately starting from the well and here turns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.

37

A man is employed to count Rs.10710. He counts at there of Rs.180 per minute for half an hour. After this, he counts at the rate of Rs.3 less every minute than the preceding minute. Find the time taken by him to count the entire amount.

38

A piece of equipment cost a certain factory Rs.600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and soon. What will be its value at the end of 10 years, all percentages applying to the original cost?

39

A sum of Rs.700 is to be used to gives even cash prizes to students of a school for their overall academic performance. If each prize is Rs.20 less than its preceding prize, find the value of each prize.

40

In an A.P. the first term is 8, nth term is 33 and the sum to first n terms is 123. Find n and d, the common differences.

41

In an A.P., the first term is 22, nth term is -11 and the sum to first n terms is 66. Find n and d, the common difference.

42

If the sum of the first n terms of an A.P. is 4n – n2, which is the first term? What is the sum of first two terms? What is the second term? Similarly, find the third, the tenth and the nth terms.

43

The first and the last term of an A.P. are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?

44

In an A.P., the first term is 2, the last term is 29 and the sum of the terms is 155. Find the common difference of the A.P.

45

In an A.P., the sum of first terms is – 150 and the sum of its next tem is-550. Find the A.P.

46

Sum of the first 14 terms of an A.P. is 1505 and its first term is 10. Find its 25th term.

47

The sum of first n terms of an A.P. is . If its mth term is 168, find the value of m. Also, find the 20th term of this A.P.

48

The sum of first q terms of an A.P. is 63q – 3q2. If its pth term is-60, find the value of p. Also, find the 11th term of this A.P.

49

The sum of first m terms of an A.P. is 4 m2 - m. If its nth term is 107, find the value of n. Also, find the 21st term of this A.P.

50

The nth term of an A.P. is given by (-4n+15). Find the sum of first 20 terms of this A.P.

51

Find the number of terms of the A.P. If 1 is added to each term of this A.P., then find the sum of all terms of the A.P. thus obtained.

52

The sum of first n terms of an A.P. is 3n2 + 4n. Find the 25th term of this A.P.

53

In a school, students decide to plant trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be double of the class in which they are studying. If there are 1 to 12 classes in the school and each class has two sections, find how many trees were planted by the students.

54

The sum of first seven terms of an A.P. is 182. If its 4th and the 17th terms are in the ratio 1:5, find the A.P.

55

The sum of the first n terms of an A.P. is 3n2 + 6n. Find the nth term of this A.P.

56

The sum of the first 7 terms of an A.P. is 63 and the sum of its next 7 terms is 161. Find the 28th term of this A.P.

57

If demotes the sum of the first n terms of an A.P., prove that S30= 3 (S20 – S10) .

58

The sum of first n terms of an A.P. is 5n – n2. Find the nth term of this A.P.

59

The sum of the first n terms of an A.P. is 4n2 + 2n. Find the nth term of this A.P.

60

If the 10th term of an A.P. is 21 and the sum of its first ten terms is 120, find its nth term.

61

Ram kali would need Rs.1800 for admission fee and books etc., for her daughter to start going to school from next year. She saved Rs.50 in the first month of this year and increased her monthly saving by Rs.20. After a year, how much money will she save? Will she be able to fulfill her dream of sending her daughter to school?

[CBSE2005,2014]

62

The first and the last terms of an A.P. are 5 and 45 respectively. If the sum of all its terms is 400, find its common difference.

63

The first and the last terms of an AP are 7 and 49 respectively. If sum of all its terms is 420, find its common difference.

64

If demotes the sum of first n terms of an A.P., prove that S12 = 3 (s8 – S4) .

65

If the sum of first n terms of an A.P. is (3n2 + 7n), then find its nth term. Hence write its 20th term.

66

The sum of first 9 terms of an A.P. is 162. The ratio of its 6th term to its 13th term is 1:2. Find the first and 15th term of the A.P.

CCE - Formative Assessment

1

Define an arithmetic progression.

2

Write the common difference of an A.P. whose nth term is an = 3n + 7.

3

Which tem of the sequence 114, 109, 104, .... is the first negative term?

4

Write the value of a30 – a10 for the A.P. 4, 9, 14, 19, ……….

5

Write 5th term from the end of the A.P. 3, 5, 7, 9, ...., 201.

6

Write the value of x for which 2x, x + 10 and 3x + 2 are in A.P.

7

Write the nth term of an A.P. the sum of whose n terms is Sn.

8

Write the sum of first n odd natural numbers.

9

Write the sum of first n even natural numbers.

10

1f the sum of n terms of an A.P. is Sn = 3n2 + 5n. Write its common difference.

11

Write the expression for the common difference of an A.P. whose first term is a and nth term is b.

12

The first term of an A.P. is p and its common difference is q. Find its 10th term. [CBSE 2008]

13

For what values of p are 2p + 1, 13, 5p – 3 are three consecutive terms of an A.P.?

14

If , a, 2 are three consecutive terms of an A.P., then find the value of a.

15

If the sum of first p term of an A.P. is ap2 + bp, find its common difference.

1

If 7th and 13th terms of an A.P. be 34 and 64 respectively, then its 18th term is

2

If the sum of P terms of an A.P. is q and the sum of q terms is p, then the sum of p + q terms will be

3

If the sum of n terms of an A.P. be 3n2 + n and its common difference is 6, then its first term is

4

The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be

5

If the sum of n terms of an A.P. is 3n2 + 5n then which of its terms is 164?

6

If the sum of n terms of an A.P. is 2n2 + 5n, then its nth term is

7

If the sum of three consecutive terms of an increasing A.P. is 51 and the product of the first and third of these terms is 273, then the third term is

8

If four numbers in A.P. are such that their sum is 50 and the greatest number is 4 times the least, then the numbers are

9

Let Sn denote the sum of n terms of an A.P. whose first term is a. If the common difference d is given by d = Sn – kSn–1 + Sn–2, then k =

10

The first and last term of an A.P. are a and l respectively. If S is the sum of all the terms of the A.P. and the common difference is given by , then k=

11

If the sum of first n even natural numbers is equal to k times the sum of first n odd natural numbers, then k =

12

If the first, second and last term of an A.P. are a, b and 2a respectively, its sum is

13

If S1 is the sum of an arithmetic progression of 'n' odd number of terms and S2 the sum of the terms of the series in odd places, then =

14

If in an A.P., Sn = n2p and Sm = m2p, where Sr denotes the sum of r terms of the A.P., then Sp is equal to

15

If Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn, then S3n : Sn is equal to

16

In an AP, Sp = q, Sq = p and Sr denotes the sum of first r terms. Then, Sp + q is equal to

17

If Sr denotes the sum of the first r terms of an A.P. Then, S3n: (S2n — Sn) is

18

If the first term of an A.P. is 2 and common difference is 4, then the sum of its 40 terms is

19

The number of terms of the A.P. 3, 7, 11, 15, ... to be taken so that the sum is 406 is

20

Sum of n terms of the series is

21

The 9th term of an A.P. is 449 and 449th term is 9. The term which is equal to zero is

22

If are in A.P. Then, x =

23

The nth term of an A.P., the sum of whose n terms is Sn, is

24

The common difference of an A.P., the sum of whose n terms is Sn, is

25

If the sums of n terms of two arithmetic progressions are in the ratio , then their nth terms are in the ratio

26

If Sn, denote the sum of n terms of an A.P. with first term a and common difference d such that , is independent of x, then

27

If the first term of an A.P. is a and nth term is b, then its common difference is

28

The sum of first n odd natural numbers is

29

Two A.P.'s have the same common difference. The first term of one of these is 8 and that of the other is 3. The difference between their 30th terms is

30

If 18, a, b, –3 are in A.P., the a + b =

31

The sum of n terms of two A.P.'s are in the ratio 5n + 9 : 9n + 6. Then, the ratio of their 18th term is

32

If , then n =

33

The sum of n terms of an A.P. is 3n2 + 5n, then 164 is its

34

If the nth term of an A.P. is 2n + 1, then the sum of first n terms of the A.P. is

35

If 18th and 11th term of an A.P. are in the ratio 3 : 2, then its 21st and 5th terms are in the ratio

36

The sum of first 20 odd natural numbers is

37

The common difference of the A.P. is , is

38

The common difference of the A.P. is

39

The common difference of the A.P. is

40

If k, 2k –1 and 2k + 1 are three consecutive terms of an AP, the value of k is