RS Agarwal Solution | Class 10th | Chapter-5 | Arithmetic Progression | Edugrown

Exercise Ex. 5B

Question 9

The sum of three consecutive terms of an AP is 21 and the sum of the squares of these terms is 165. Find these terms.Solution 9

Let the required numbers be (a – d), a, (a + d)

Sum of these numbers = (a – d) + a + (a + d) = 3a

sum of these squares = 

Sum of three numbers = 21, sum of squares of these numbers = 165

3a = 21

a = 7

Thus, a = 7 and d = 

Hence, the required numbers are (4, 7, 10) or (10, 7, 4)Question 1

Determine k so that (3k – 2), (4k – 6) and (k + 2) are three consecutive terms of an AP.Solution 1

Question 2

Find the value of x for which the numbers (5x + 2),(4x – 1) and (x + 2) are in AP.Solution 2

Question 3

If (3y – 1),(3y + 5) and (5y + 1) are three consecutive terms of an AP then find the value of y.Solution 3

Question 4

Find the value of x for which (x + 2), 2x, (2x + 3) are three consecutive terms of an AP.Solution 4

If are consecutive terms of an AP, then

Question 5

Solution 5

Question 6

Find three numbers in AP whose sum is 15 and product is 80.Solution 6

Let the required numbers be (a – d), a and (a + d)

Sum of these numbers = (a – d) + a + (a + d) = 3a

Product of these numbers = (a – d) × a × (a + d) = 

But sum = 15 and product = 80

Hence, the required numbers are (2, 5, 8)Question 7

The sum of three numbers in AP is 3 and their product is -35. Find the numbers.Solution 7

Let the required numbers be (a – d), a, (a + d)

Sum of these number = (a – d) + a + (a + d) = 3a

Product of these numbers = (a – d) a (a + d)

But,sum = 3 and product = – 35

Thus, a = 1 and d = 6

Hence, the required numbers are (-5, 1, 7)Question 8

Divide 24 into three parts such that they are in AP and their product is 440.Solution 8

Let the required number be (a – d), a and (a + d)

Sum of these numbers = (a – d) + a + (a + d) = 3a

Product of these numbers = (a – d) x a x (a + d) 

But sum = 24 and product = 440

Thus, a = 8 and d = 3

Hence the required numbers are (5, 8, 11)Question 10

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

Let the required angles be (a – 3d)°, (a – d) °, (a + d) ° and (a + 3d) °

Common difference = (a – d) – (a- 3d) = a – d – a + 3d = 2d

Common difference = 10°

2d = 10° = d = 5°

Sum of four angles of quadrilateral = 360°

First angle = (a – 3d)° = (90 – 3 × 5) ° = 75°

Second angle = (a – d)° = (90 – 5) ° = 85°

Third angle = (a + d)° = (90 + 5°) = 95°

Fourth angle = (a + 3d)° = (90 + 3 × 5)° = 105°Question 11

Find four numbers in AP whose sum is 28 and the sum of whose squares is 216.Solution 11

Let the required number be (a – 3d), (a – d), (a + d) and (a + 3d)

Sum of these numbers = (a – 3d) + (a – d)+ (a + d) + (a + 3d)

4a = 28  a = 7

Sum of the squares of these numbers

Hence, the required numbers (4, 6, 8, 10)Question 12

Divide 32 into four parts which are the four terms of an AP such that the product of the first and the fourth terms is to the product of the second and the third terms as 7 : 15.Solution 12

Question 13

The sum of first three terms of an AP is 48. If the product of first and second terms exceeds 4 times the third term by 12. Find the AP.Solution 13

Exercise Ex. 5C

Question 26

The 12^th term of an AP is -13 and the sum of its first four terms is 24. Find the sum of its first 10 terms.Solution 26

Question 1(i)

Find the sum of each of the following APs:

2, 7, 12, 17, ….. to 19 terms.Solution 1(i)

Question 1(ii)

Find the sum of each of the following APs:

9, 7, 5, 3, ….. to 14 terms.Solution 1(ii)

Question 1(iii)

Find the sum of each of the following APs:

-37, -33, -29, …. to 12 terms.Solution 1(iii)

Question 1(iv)

Find the sum of each of the following APs:

Solution 1(iv)

Question 1(v)

Find the sum of each of the following APs:

0.6, 1.7, 2.8, ….. to 100 terms.Solution 1(v)

Question 2(i)

Find the sum of each of the following arithmetic series:

Solution 2(i)

Question 2(ii)

Find the sum of each of the following arithmetic series:

Solution 2(ii)

Question 2(iii)

Find the sum of each of the following arithmetic series:

Solution 2(iii)

Question 3

Find the sum of first n terms of an AP whose n^th term is (5 – 6n). Hence, find the sum of its first 20 terms.Solution 3

Question 4

Solution 4

Question 5

If the sum of the first n terms of an AP is given by S_n = (3n² – n), find its (i) nth term, (ii) first term and (iii) common difference.Solution 5

(i)The nth term is given by

(ii)Putting n = 1 in (1) , we get

(iii)Putting n = 2 in (1), we get = 8

Question 6(i)

The sum of n terms of an AP is . Find its 20^th term.Solution 6(i)

It is given that —–(1)

Now, 20^th term

=(sum of first 20 term) – (sum of first 19 terms)

Putting = 20 in (1) we get

Putting n= 19 in (1), we get

Hence, the 20^thterm is 99Question 6(ii)

Solution 6(ii)

Question 8

How many terms of the AP 21, 18, 15, … must be added to get the sum 0?Solution 8

Here a = 21, d = (18 – 21) = -3

Let the required number of terms be n, then

sum of first 15 terms = 0Question 9

How many terms of the AP 9, 17, 25, …. must be taken so that their sum is 636?Solution 9

Question 10

How many terms of the AP 63, 60, 57, 54, … must be taken so that their sum is 693? Explain the double answer.Solution 10

Question 11

Solution 11

Question 12

Write the next term of the APSolution 12

The term AP is 

Question 13

Find the sum of all natural numbers between 200 and 400 which are divisible by 7.Solution 13

Question 14

Find the sum of first forty positive integers divisible by 6.Solution 14

Question 15

The nth term of an AP is (7 – 4n). Find its common difference.Solution 15

Question 16

Find the sum of all multiples of 9 lying between 300 and 700.Solution 16

All numbers between 300 and 700 that are multiples of 9 are 306, 315, 324, 333, …, 693

This is an AP in which a = 306, d = (315 – 306) = 9, l = 693

Let the number of these terms be n, then

Question 17

The nth term of an AP is (3n + 5). Find its common difference.Solution 17

Thus, common difference = 3Question 18

Write the next term of the APSolution 18

The given AP is 

Common difference d = 

Term next to Question 19

Find the sum of the following:

Solution 19

The given AP is 

First term 

Common difference d = 

Sum of n terms = 

Question 20

Solution 20

Question 21

In an AP the first term is 2, the last term is 29 and sum of the terms is 155. Find the common difference of the AP.Solution 21

First term ‘a’ of an AP = 2

The last term l = 29

common difference = 3Question 22

In an AP, the first term is -4, the last term is 29 and the sum of all its terms is 150. Find its common difference.Solution 22

Question 23

The first and the last terms of an AP are 17 and 350 respectively. If the common difference is 9, how many terms are there and what is their sum?Solution 23

Question 24

The first and the last terms of an AP are 5 and 45 respectively. If the sum of all its terms is 400, find the common difference and the number of terms.Solution 24

Question 25

In an AP the first term is 22, n^th term is -11 and sum to first n^th terms is 66. Find n and d, the common difference.Solution 25

First term of an AP, a = 22

Last term = n^th term = – 11

Thus, n = 12, d = -3Question 27

The sum of the first 7 terms of an AP is 182. If its 4^th and 17^th terms are in the ratio 1 : 5, find the AP.Solution 27

Question 28

The sum of the first 9 terms of an AP is 81 and that of its first 20 terms is 400. Find the first term and the common difference of the AP.Solution 28

Question 29

The sum of the first 7 terms of an AP is 49 and the sum of its first 17 terms is 289. Find the sum of its first n terms.Solution 29

Question 30

Two APs have the same common difference. If the first terms of these APs be 3 and 8 respectively, find the difference between the sums of their first 50 terms.Solution 30

Question 31

The sum of first 10 terms of an AP is -150 and the sum of its next 10 terms is -550. Find the AP.Solution 31

Question 32

The 13^th term of an AP is 4 times its 3^rd term. If its 5^th term is 16, find the sum of its first 10 terms.Solution 32

Question 33

The 16^th term of an AP is 5 times its 3^rd term. If its 10^th term is 41, find the sum of its first 15 terms.Solution 33

Question 34(i)

An AP 5, 12, 19, … has 50 terms. Find its last term. Hence, find the sum of its last 15 terms.Solution 34(i)

Question 34(ii)

An AP 8, 10, 12, … has 60 terms. Find its last term. Hence, find the sum of its last 10 terms.Solution 34(ii)

Question 36

The sum of the 4^th and the 8^th terms of an AP is 24 and the sum of its 6^th and 10^th terms is 44. Find the sum of its first 10 terms.Solution 36

Question 37

The sum of first m terms of an AP is (4m² – m).If its nth term is 107, find the value of n. Also ,find the 21^st term of this AP.Solution 37

Question 38

The sum of first q terms of an AP is (63q -3q²). If its pth term is -60, find the value of p. Also , find the 11^th term of its AP.Solution 38

Question 39

Find the number of terms of the AP -12, -9, -6, …, 21. If 1 is added to each term of this AP then find the sum of all terms of the AP thus obtained.Solution 39

Question 40

Sum of the first 14 terms of an AP is 1505 and its first term is 10. Find its 25^th term.Solution 40

Question 41

Find the sum of first 51 terms of an AP whose second and third terms are 14 and 18 respectively.Solution 41

Question 42

In a school, students decided 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 students. Which value is shown in the question?Solution 42

Question 43

In a potato race, a bucket is placed at the starting point, which is 5 m from the first potato, and the other potatoes are placed 3 m apart in a straight line. There are 10 potatoes in the line. A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and he continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

Solution 43

Question 44

There are 25 trees at equal distances of 5 m in a line with a water tank, the distance of the water tank from the nearest tree being 10 m. A gardener waters all the trees separately, starting from the water tank and returning back to the water tank after watering each tree to get water for the next. Find the total distance covered by the gardener in order to water all the trees.

Solution 44

Question 45

A sum of Rs. 700 is to be used to give seven 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.Solution 45

Question 46

A man saved Rs. 33000 in 10 months. In each month after the first, he saved Rs. 100 more than he did in the preceding month. How much did he save in the first month?Solution 46

Question 47

A man arranges to pay off a debt of Rs. 36000 by 40 monthly installments which form an 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.Solution 47

Question 48

A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows:

Rs. 200 for the first day, Rs. 250 for the second day, Rs. 300 for the third day, etc., the penalty for each succeeding day being Rs. 50 more than for the preceding day. How much money the 1 contractor has to pay as penalty, if he has delayed the work by 30 days?Solution 48

Exercise MCQ

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

If 4, x₁, x₂, x₃, 28 are in AP then x₃= ?

(a) 19

(b) 23

(c) 22

(d) cannot be determinedSolution 4

Question 5

If the nth term of an AP is (2n + 1) then the sum of its first three terms is

(a) 6n + 3

(b) 15

(c) 12

(d) 21Solution 5

Question 6

The sum of first n terms of an AP is (3n² + 6n). The common difference of the AP is

(a) 6

(b) 9

(c) 15

(d) -3Solution 6

Question 7

The sum of first n terms of an AP is (5n – n²). The nth term of the AP is

(a) (5 – 2n)

(b) (6 – 2n)

(c) (2n – 5)

(d) (2n – 6)Solution 7

Question 8

The sum of first n terms of an AP is (4n² + 2n). The nth term of this AP is

(a) (6n – 2)

(b) (7n – 3)

(c) (8n – 2)

(d) (8n + 2)Solution 8

Question 9

The 7th term of an AP is -1 and its 16th term is 17. The nth term of AP is

(a) (3n + 8)

(b) (4n – 7)

(c) (15 – 2n)

(d) (2n -15)Solution 9

Question 10

The 5th term of an AP is -3 and its common difference is -4. The sum of its first 10 terms is

(a) 50

(b) -50

(c) 30

(d) -30Solution 10

Question 11

The 5th term of an AP is 20 and the sum of its 7th and 11th terms is 64. The common difference of the AP is

(a) 4

(b) 5

(c) 3

(d) 2Solution 11

Question 12

The 13th term of an AP is 4 times its 3rd term. If its 5th term is 16 then the sum of its first ten terms is

(a) 150

(b) 175

(c) 160

(d) 135Solution 12

Question 13

An AP 5, 12, 19, … has 50 terms. Its last term is

(a) 343

(b) 353

(c) 348

(d) 362Solution 13

Question 14

The sum of first 20 odd natural numbers is

(a) 100

(b) 210

(c) 400

(d) 420Solution 14

Question 15

The sum of first 40 positive integers divisible by 6 is

(a) 2460

(b) 3640

(c) 4920

(d) 4860Solution 15

Question 16

How many two-digit numbers are divisible by 3?

(a) 25

(b) 30

(c) 32

(d) 36Solution 16

Question 17

How many three-digit numbers are divisible by 9?

(a) 86

(b) 90

(c) 96

(d) 100Solution 17

Question 18

What is the common difference of an AP in which a₁₈ – a₁₄ = 32?

(a) 8

(b) -8

(c) 4

(d) -4Solution 18

Question 19

If a_n denotes the nth term of the AP 3, 8, 13, 18, … then what is the value of (a₃₀ -a₂₀)?

(a) 40

(b) 36

(c) 50

(d) 56Solution 19

Question 20

Which term of the AP 72, 63, 54, … is 0?

(a) 8^th

(b) 9^th

(c) 10^th

(d) 11^thSolution 20

Question 21

Which term of the AP 25, 20, 15, … is the first negative term?

(a) 10^th

(b) 9^th

(c) 8^th

(d) 7^thSolution 21

Question 22

Which term of the AP 21, 42, 63, 84, … is 210?

(a) 9^th 

(b) 10^th 

(c) 11^th 

(d) 12^thSolution 22

Question 23

What is 20th term from the end of the AP 3, 8, 13, …, 253?

(a) 163

(b) 158

(c) 153

(d) 148Solution 23

Question 24

(5 + 13 +21+… + 181) =?

(a) 2476

(b) 2337

(c) 2219

(d) 2139Solution 24

Question 25

The sum of first 16 terms of the AP 10, 6, 2, … is

(a) 320

(b) -320

(c) -352

 (d) -400Solution 25

Question 26

How many terms of the AP 3, 7, 11, 15, … will make the sum 406?

(a) 10

(b) 12

(c) 14

(d) 20Solution 26

Question 27

The 2^nd term of an AP is 13 and its 5^th term is 25. What is its 17^th term?

(a) 69

(b) 73

(c) 77

(d) 81Solution 27

Question 28

The 17^th term of an AP exceeds its 10^th term by 21. The common difference of the AP is

(a) 3

(b) 2

(c) -3

(d) -2Solution 28

Question 29

The 8^th term of an AP is 17 and its 14^th term is 29. The common difference of the AP is

(a) 3

(b) 2

(c) 5

(d) -2Solution 29

Question 30

The 7^th term of an AP is 4 and its common difference is -4. What is its first term?

(a) 16

(b) 20

(c) 24

(d) 28Solution 30

Exercise Ex. 5A

Question 1(i)

Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each.

9, 15, 21, 27, …..Solution 1(i)

Question 1(ii)

Show that each of the progressions given below is an AP. Find the first term, common difference and next term of each.

11, 6, 1, -4, ….Solution 1(ii)

Question 1(iii)

Solution 1(iii)

Question 1(iv)

Solution 1(iv)

Question 1(v)

Solution 1(v)

Question 2(i)

Find

The 20^th term of the AP 9, 13, 17, 21, ….Solution 2(i)

Question 2(ii)

Find

The 35^th term of the AP 20, 17, 14, 11, ….Solution 2(ii)

Question 2(iii)

Solution 2(iii)

Question 2(iv)

Solution 2(iv)

Question 2(v)

Find

The 15^th term of the AP -40, -15, 10, 35, ….Solution 2(v)

Question 3(i)

Find the 37^th term of the AP Solution 3(i)

The given AP is 

First term = 6, common difference = 

a = 6, d = 

The n^th term is given by

Hence, 37^th term is 69
Question 3(ii)

Find the 25^th term of the AP Solution 3(ii)

The given AP is 

The first term = 5,

common difference = 

a = 5, 

The n^th term is given by

Hence the 25^th term is – 7Question 5(i)

Find the nth term of each of the following APs :

5, 11, 17, 23, …..Solution 5(i)

Question 5(ii)

Find the nth term of each of the following APs :

16, 9, 2, -5, …..Solution 5(ii)

Question 6

If the nth term of a progression is (4n – 10), show that it is an A.P. Find its (i) first term, (ii) common difference, and (iii) 16^th term.Solution 6

(i)First term = -6

(ii)Common difference 

(iii)16^th term = where a = -6 and d = 4

       = (-6 + 15 4) = 54Question 7

How many terms are there in the AP 6, 10, 14, 18, …, 174?Solution 7

In the given AP, we have a = 6 and d = (10 – 6) = 4

Suppose there are n terms in the given AP, then

Hence there are 43 terms in the given APQuestion 8

How many terms are there in the AP 41, 38, 35, …8?Solution 8

In the given AP we have a = 41 and d = 38 – 41 = – 3

Suppose there are n terms in AP, then

Hence there are 12 terms in the given APQuestion 9

Solution 9

Question 10

Which term of the AP 3, 8, 13, 18, … is 88?Solution 10

In the given AP, we have a = 3 and d = 8 – 3 = 5

Suppose there are n terms in given AP, then

Hence, the 18^th term of given AP is 88Question 11

Which term of the AP 72, 68, 64, 60, … is 0?Solution 11

In the given AP, we have a = 72 and d = 68 – 72 = – 4

Suppose there are n terms in given AP, we have

Hence, the 19^th term in the given AP is 0Question 12

Which term of the AP is 3?Solution 12

In the given AP, we have 

Suppose there are n terms in given AP, we have

Then,

Thus, 14^th term in the given AP is 3Question 13

Which term of the AP 21, 18, 15, … is -81?Solution 13

Question 15

Which term of the AP 5, 15, 25, … will be 130 more than its 31^st term?Solution 15

The given AP is 5, 15, 25….

a = 5, d = 15 – 5 = 10

Thus, the required term is 44^thQuestion 16

If the 10^th term of an AP is 52 and the 17^th term is 20 more than the 13^th term, find the AP.Solution 16

In the given AP let the first term = a,

And common difference = d

So the required AP is 7, 12, 17, 22….Question 17

Find the middle term of the AP 6, 13, 20, …, 216.Solution 17

Question 18

Find the middle term of the AP 10, 7, 4, …, (-62).Solution 18

Question 19

Solution 19

Question 20

Find the 8^th term from the end of the AP 7, 10, 13, …, 184.Solution 20

Here a = 7, d = (10 – 7) = 3, l = 184

And n = 8

Hence, the 8^th term from the end is 163Question 21

Find the 6^th term from the end of the AP 17, 14, 11, …(-40).Solution 21

Here a = 17, d = (14 – 17) = -3, l = -40

And n = 6

Now, n^th term from the end = [l – (n – 1)d]

Hence, the 6^th term from the end is – 25Question 22

Is 184 a term of the AP 3, 7, 11, 15, …?Solution 22

Question 23

Is-150 a term of the AP 11, 8, 5, 2, …?Solution 23

Question 24

Which term of the AP 121, 117, 113, … is its first negative term?Solution 24

Question 25

Solution 25

Question 26

The 7^th term of an AP is -4 and its 13^th term is -16. Find the AP.Solution 26

In the given AP, let the first term = a common difference = d

So the required AP is 8, 6, 4, 2, 0……Question 27

The 4^th term of an AP is zero. Prove that its 25^th term is triple its 11^th term.Solution 27

Let the first term of given AP = a and common difference = d

Hence 25^th term is triple its 11^th termQuestion 29

The 4^th term of an AP is 11. The sum of the 5^th and 7^th terms of this AP is 34. Find its common difference.Solution 29

Question 30

The 9^th term of an AP is -32 and the sum of its 11^th and 13^th terms is -94. Find the common difference of the AP.Solution 30

Question 31

Determine the nth term of the AP whose 7th term is -1 and 16th term is 17.Solution 31

Question 32

If 4 times the 4th term of an AP is equal to 18 times its 18th term then find its 22nd term.Solution 32

Question 33

If 10 times the 10^th term of an AP is equal to 15 times the 15^th term, show that its 25^th term is zero.Solution 33

Let a be the first term and d be the common difference

Question 34

Find the common difference of an AP whose first term is 5 and the sum of its first four terms is half the sum of the next four terms.Solution 34

Question 35

The sum of the 2nd and the 7th terms of an AP is 30. If its 15th term is 1 less than twice its 8th term, find the AP.Solution 35

Question 36

For what value of n, the n^th terms of the APs 63, 65, 67, … and 3, 10, 17, … are equal?Solution 36

First AP is 63, 65, 67….

First term = 63, common difference = 65 – 63 = 2

n^thterm = 63 + (n – 1) 2 = 63 + 2n – 2 = 2n + 61

Second AP is 3, 10, 17 ….

First term = 3, common difference = 10 – 3 = 7

n^th term = 3 + (n – 1) 7 = 3 + 7n – 7 = 7n – 4

The two n^th terms are equal

2n + 61 = 7n – 4 or 5n = 61 + 4 = 65

Question 37

The 17th term of AP is 5 more than twice its 8th term. If the 11th term of the AP is 43, find its nth term.Solution 37

Question 38

The 24th term of an AP is twice its 10th term. Show that its 72nd term is 4 times its 15th term.Solution 38

Question 39

The 19th term of an AP is equal to 3 times its 6th term. If its 9th term is 19, find the AP.Solution 39

Question 40

If the p^th term of an AP is q and its q^th term is p, then show that its (p + q)^th term is zero.Solution 40

Let a be the first term and d be the common difference

p^th term = a +(p – 1)d = q(given)—–(1)

q^th term = a +(q – 1) d = p(given)—–(2)

subtracting (2) from (1)

(p – q)d = q – p

(p – q)d = -(p – q)

 d = -1

Putting d = -1 in (1)

Question 41

The first and last terms of an AP are a and l respectively. Show that the sum of the n^th term from the beginning and the n^th term from the end is (a + l).Solution 41

Let a be the first term and d be the common difference

n^th term from the beginning = a + (n – 1)d—–(1)

n^th term from end= l – (n – 1)d —-(2)

adding (1) and (2),

sum of the n^th term from the beginning and n^th term from the end = [a + (n – 1)d] + [l – (n – 1)d] = a + lQuestion 42

How many two-digit numbers are divisible by 6?Solution 42

Question 43

How many two-digit numbers are divisible by 3?Solution 43

Question 44

How many three-digit numbers are divisible by 9?Solution 44

Question 45

How many numbers are there between 101 and 999, which are divisible by both 2 and 5?Solution 45

Question 46

In a flower bed, there are 43 rose plants in the first row, 41 in the second, 39 in the third, and so on. There are 11 rose plants in the last row. How many rows are there in the flower bed?Solution 46

Number of rose plants in first, second, third rows…. are 43, 41, 39… respectively.

There are 11 rose plants in the last row

So, it is an AP . viz. 43, 41, 39 …. 11

a = 43, d = 41 – 43 = -2, l = 11

Let n^th term be the last term

Hence, there are 17 rows in the flower bed.Question 47

A sum of Rs. 2800 is to be used to award four prizes. If each prize after the first is Rs. 200 less than the preceding prize, find the value of each of the prizes.Solution 47

Exercise Ex. 5D

Question 1

The first three terms of an AP are respectively (3y – 1), (3y + 5) and (5y + 1), find the value of y.Solution 1

Question 2

If k, (2k – 1) and (2k + 1) are the three successive terms of an AP, find the value of k.Solution 2

Question 3

If 18, a, (b – 3) are in AP, then find the value of (2a – b).Solution 3

Question 4

If the numbers a, 9, b, 25 form an AP, find a and b.Solution 4

Question 5

If the numbers (2n – 1), (3n + 2) and (6n – 1) are in AP, find the value of n and the numbers.Solution 5

Question 6

How many three-digit natural numbers are divisible by 7?Solution 6

Question 7

How many three-digit natural numbers are divisible by 9?Solution 7

Question 8

If the sum of first m terms of an AP is (2m² + 3m) then what is its second term?Solution 8

Question 9

What is the sum of first n terms of the AP a, 3a, 5a, ….Solution 9

Question 10

What is the 5th term from the end of the AP 2, 7, 12, …, 47?Solution 10

Question 11

If a_n denotes the nth term of the AP 2, 7, 12, 17, …, find the value of (a₃₀ – a₂₀).Solution 11

Question 12

Find the sum of all three-digit natural numbers which are divisible by 13.Solution 12

All three digit natural numbers divisible by 13 are 104, 117, 130, 143,…, 988

This is an AP in which a = 104, d = (117 – 104) = 13, l = 988

Question 13

Find the sum of first 15 multiples of 8.Solution 13

First 15 multiples of 8 are 8, 16, 24, … to 15^th term

Question 14

Find the sum of all odd numbers between 0 and 50.Solution 14

Odd natural numbers between 0 and 50 are 1, 3, 5, … 49

a = 1, d = 3 – 1= 2, l = 49

Let the number of terms be n

Question 15

Find the sum of first hundred even natural numbers which are divisible by 5.Solution 15

First 100 even natural numbers divisible by 5 are

10, 20, 30, … to 100 term

First term of AP = 10

Common difference d = 20 – 10 = 10

Number of terms = n = 100

Question 16

Which term of the AP 21, 18, 15, … is zero?Solution 16

The given AP is 21, 18, 15, ….

First term = 21, common difference = 18 – 21= – 3

Let n^th term be zero

a + (n – 1)d = 0or 21 + (n – 1)(-3) = 0

21 – 3n + 3 = 0

 3n = 24

or 

Hence, 8^th term of given series is 0Question 17

Find the sum of first n natural numbers.Solution 17

Sum of n natural numbers = 1 + 2 + 3 + … + n

Here a = 1, d = 2 – 1 = 1

Question 18

Find the sum of first n even natural numbers.Solution 18

Sum of even natural numbers = 2 + 4 + 6 + … to n terms

a = 2, d = 4 – 2 = 2

Question 19

The first term of an AP is p and its common difference is q. Find its 10^th term.Solution 19

First term of AP = a = p

Common difference = d = q

n^th term = a + (n 1)d

10^th term = p + (10 1)q

               = p + 9qQuestion 20

If are three consecutive terms of an AP, find the value of a.Solution 20

Question 21

If (2p + 1), 13, (5p – 3) are in AP, find the value of p.Solution 21

Question 22

If (2p – 1), 7, 3p are in AP, find the value of p.Solution 22

Question 23

If the sum of first p terms of an AP is (ap² + bp), find its common difference.Solution 23

Question 24

If the sum of first n terms is (3n² + 5n), find its common difference.Solution 24

Question 25

Find an AP whose 4th term is 9 and the sum of its 6th and 13th terms is 40.Solution 25