Class 10 Maths · Chapter 5

Arithmetic Progressions

137 Class 10 Maths questions on Arithmetic Progressions from the NCERT textbook, NCERT Exemplar and CBSE board papers 2026, each with its answer and a step-by-step solution.

CBSE Class 10 Maths previous year papers, with answers →

Questions · page 4 of 7

1. Q61 Exemplar · Exercise 5.1
   In ​an ​AP ​if $a=1$, $a_n=20$ and $S_n=399$, ​then $n$ ​is

   1. (a)19
   2. (b)21
   3. (c)38
   4. (d)42
   Show answer

   (c) 38

   Step-by-step solution →
2. Q62 Exemplar · Exercise 5.1
   The ​sum ​of first ​five ​multiples ​of ​3 is

   1. (a)45
   2. (b)55
   3. (c)65
   4. (d)75
   Show answer

   (a) 45

   Step-by-step solution →
3. Q63 Exemplar · Exercise 5.2
   Which of ​the following form an ​AP? ​Justify your ​answer. ​

   1. (i)$-1,-1,-1,-1,\dots$
   2. (ii)$0,2,0,2,\dots$
   3. (iii)$1,1,2,2,3,3,\dots$
   4. (iv)$11,22,33,\dots$ ​
   5. (v)$\frac{1}{2},\frac{1}{3},\frac{1}{4},\dots$ ​
   6. (vi)$2,2^2,2^3,2^4,\dots$
   7. (vii)$\sqrt{3},\sqrt{12},\sqrt{27},\sqrt{48},\dots$
   Show answer

   (i) AP ($d=0$); (ii) Not an AP; (iii) Not an AP; (iv) AP ($d=11$); (v) Not an AP; (vi) Not an AP (it is a GP); (vii) AP ($d=\sqrt{3}$).

   Step-by-step solution →
4. Q64 Exemplar · Exercise 5.2
   For ​the AP: $-3,-7,-11,\dots$, can we find directly $a_{30}-a_{20}$ without actually ​finding $a_{30}$ and $a_{20}$? ​Give reasons ​for ​your ​answer.
   Show answer

   Yes - $a_{30}-a_{20}=10d=10(-4)=-40$, obtained directly because $a$ cancels.

   Step-by-step solution →
5. Q65 Exemplar · Exercise 5.2
   Two APs have the same ​common difference. The first term of one ​AP is 2 and that of ​the other ​is ​7. The difference between their 10th terms is the same as the difference between their 21st terms, which is the ​same as the difference between any two corresponding terms. Why?
   Show answer

   Because both APs have the same common difference d, the term A_n - a_n = [7 + (n-1)d] - [2 + (n-1)d] = 5 for every n. The (n-1)d parts cancel, so the difference of corresponding terms is always 5, the difference of the first terms.

   Step-by-step solution →
6. Q66 Exemplar · Exercise 5.2
   Is $0$ a term ​of ​the ​AP: $31,28,25,\dots$? ​Justify ​your ​answer.
   Show answer

   No. Setting $34-3n=0$ gives $n=\frac{34}{3}$, which is not a positive integer, so $0$ is not a term of the AP.

   Step-by-step solution →
7. Q67 Exemplar · Exercise 5.2
   The taxi fare after each km, when the fare is Rs 15 for the ​first km and Rs ​8 ​for each additional ​km, does not form an AP as the ​total fare (in Rs) after ​each km ​is 15, 8, 8, 8, ... Is the statement ​true? Give reasons.
   Show answer

   The statement is false. The correct total-fare sequence is $15,23,31,39,\dots$, which **is** an AP with first term 15 and common difference 8.

   Step-by-step solution →
8. Q68 Exemplar · Exercise 5.2
   In which of the following situations, ​do the lists of numbers involved form an AP? ​Give reasons for your answers.

   1. (i)The fee charged from a student every month by a school for the whole session, when the monthly fee is Rs 400. ​
   2. (ii)​The fee charged every month by a school from Classes I to XII, when the monthly ​fee for Class I is Rs ​250, and it increases by Rs 50 for ​the next higher class.
   3. (iii)The amount of money in the account of Varun at the end of every year when Rs 1000 is deposited at simple interest of 10% per annum.
   4. (iv)The number of bacteria ​in a certain food item after each second, when they double in every second.

   1. (i)Rs 400 monthly fee each month
   2. (ii)Class I fee Rs 250, increases Rs 50 per class
   3. (iii)Rs 1000 at 10% simple interest, yearly amount
   4. (iv)Bacteria doubling every second
   Show answer

   (i), (ii) And (iii) form APs (common differences $0$, $50$, $100$); (iv) Does not - it is a GP with ratio $2$.

   Step-by-step solution →
9. Q69 Exemplar · Exercise 5.2
   Justify ​whether ​it ​is true to ​say that the following ​are ​the $n^{\text{th}}$ terms of an ​AP.

   1. (i)$2n-3$
   2. (ii)$3n^2+5$
   3. (iii)$1+n+n^2$
   Show answer

   (i) $2n-3$ **is** an AP (common difference 2). (ii) $3n^2+5$ is **not** an AP. (iii) $1+n+n^2$ is **not** an AP.

   Step-by-step solution →
10. Q70 Exemplar · Exercise 5.3
    Match ​the APs given ​in column ​A ​with suitable ​common ​differences given in column ​B.
    Show answer

    Matching: A1-B4 ($d=-4$), A2-B5 ($d=2$), A3-B1 ($d=2/3$), A4-B2 ($d=-5$).

    Step-by-step solution →
11. Q71 Exemplar · Exercise 5.3
    Verify ​that each of the ​following ​is an ​AP, and then ​write ​its next three terms. ​

    1. (i)$0,\frac{1}{4},\frac{1}{2},\frac{3}{4},\dots$
    2. (ii)$5,\frac{14}{3},\frac{13}{3},4,\dots$
    3. (iii)$\sqrt{3},2\sqrt{3},3\sqrt{3},\dots$
    4. (iv)$a+b,(a+1)+b,(a+1)+(b+1),\dots$
    5. (v)$a,2a+1,3a+2,4a+3,\dots$
    Show answer

    All five lists are APs. Next three terms: (i) $1,\frac{5}{4},\frac{3}{2}$; (ii) $\frac{11}{3},\frac{10}{3},3$; (iii) $4\sqrt{3},5\sqrt{3},6\sqrt{3}$; (iv) $a+b+3,a+b+4,a+b+5$; (v) $5a+4,6a+5,7a+6$.

    Step-by-step solution →
12. Q72 Exemplar · Exercise 5.3
    Write the first three ​terms ​of ​the ​APs ​when $a$ and $d$ ​are as given below:

    1. (i)$a=\frac{1}{2},d=-\frac{1}{6}$ ​
    2. (ii)$a=-5,d=-3$
    3. (iii)$a=\sqrt{2},d=\frac{1}{\sqrt{2}}$
    Show answer

    (i) $\frac{1}{2},\frac{1}{3},\frac{1}{6}$ (ii) $-5,-8,-11$ (iii) $\sqrt{2},\frac{3}{\sqrt{2}},2\sqrt{2}$.

    Step-by-step solution →
13. Q73 Exemplar · Exercise 5.3
    Find $a$, $b$ and $c$ such ​that ​the ​following ​numbers ​are ​in ​AP: $a,7,b,23,c$.
    Show answer

    $a=-1$, $b=15$, $c=31$.

    Step-by-step solution →
14. Q74 Exemplar · Exercise 5.3
    Determine the ​AP ​whose fifth term is 19 ​and the difference ​of the eighth ​term from the thirteenth ​term is 20.
    Show answer

    The AP is 3, 7, 11, 15, 19, ... With first term 3 and common difference 4.

    Step-by-step solution →
15. Q75 Exemplar · Exercise 5.3
    The 26th, 11th ​and the last term of ​an ​AP are 0, 3 and $-\frac{1}{5}$, respectively. Find the common ​difference and the ​number of ​terms.
    Show answer

    Common difference $d=-\frac{1}{5}$ and the number of terms $n=27$ (with first term $a=5$).

    Step-by-step solution →
16. Q76 Exemplar · Exercise 5.3
    The sum ​of the ​5th and the 7th terms of an ​AP ​is 52 ​and the 10th ​term is 46. Find the AP.
    Show answer

    The AP is $1,6,11,16,21,\dots$ with first term $a=1$ and common difference $d=5$.

    Step-by-step solution →
17. Q77 Exemplar · Exercise 5.3
    Find ​the ​20th term ​of the AP whose 7th term is ​24 less than ​the 11th ​term, first term ​being 12.
    Show answer

    The 20th term is $a_{20}=126$.

    Step-by-step solution →
18. Q78 Exemplar · Exercise 5.3
    If the $9^{th}$ term ​of an AP ​is ​zero, prove ​that ​its $29^{th}$ ​term ​is ​twice its $19^{th}$ term.
    Show answer

    Proved: with $a_9=0$ we get $a=-8d$, so $a_{19}=10d$ and $a_{29}=20d=2a_{19}$.

    Step-by-step solution →
19. Q79 Exemplar · Exercise 5.3
    Find whether ​55 ​is a term ​of the AP: 7, 10, ​13, ... or ​not. If yes, find which ​term it ​is.
    Show answer

    Yes, 55 is a term of the AP -- it is the 17th term.

    Step-by-step solution →
20. Q80 Exemplar · Exercise 5.3
    Determine $k$ so ​that $k^2+4k+8$, $2k^2+3k+6$, $3k^2+4k+4$ ​are ​three ​consecutive ​terms of ​an ​AP.
    Show answer

    $k=0$; the three terms become $8$, $6$, $4$ with common difference $-2$.

    Step-by-step solution →