Arithmetic Progressions: Class 10 Important Questions with Answers

Case-based practice

20 MCQs (1 mark each)

1. The common difference of 3, 1, −1, −3, … is
   (a) 2 (b) −2 (c) 3 (d) −4
2. The 10th term of 2, 7, 12, … is
   (a) 45 (b) 47 (c) 52 (d) 50
3. Which of these is an AP?
   (a) 1, 4, 9, 16 (b) 2, 4, 8, 16 (c) $\sqrt{2},\sqrt{8},\sqrt{18},\sqrt{32}$ (d) $1,\frac{1}{2},\frac{1}{3},\frac{1}{4}$
4. If $2x$, $x+10$, $3x+2$ are in AP, then $x$ is
   (a) 6 (b) 4 (c) 5 (d) 3
5. The nth term of 5, 9, 13, … is
   (a) $4n+5$ (b) $4n+1$ (c) $5n+4$ (d) $4n-1$
6. The sum of the first 20 natural numbers is
   (a) 200 (b) 210 (c) 420 (d) 190
7. If $a_n=7-3n$, the common difference is
   (a) 7 (b) 3 (c) −3 (d) 4
8. The number of terms in 7, 11, 15, …, 139 is
   (a) 33 (b) 34 (c) 35 (d) 36
9. The sum of the first $n$ odd natural numbers is
   (a) $n(n+1)$ (b) $n^2$ (c) $2n$ (d) $\frac{n(n+1)}{2}$
10. If $S_n=n^2+2n$, the first term is
    (a) 1 (b) 2 (c) 3 (d) 4
11. The 15th term from the end of 3, 5, 7, …, 81 is
    (a) 53 (b) 51 (c) 55 (d) 31
12. If $a=10$ and $d=-2$, then $a_6$ is
    (a) 0 (b) 2 (c) −2 (d) 20
13. Which term of 21, 18, 15, … is 0?
    (a) 7th (b) 8th (c) 9th (d) 6th
14. $5+10+15+\cdots +100$ equals
    (a) 1050 (b) 1000 (c) 2100 (d) 950
15. The first negative term of 50, 46, 42, … is the
    (a) 13th (b) 14th (c) 15th (d) 12th
16. If $a_3=5$ and $a_7=9$, then $d$ is
    (a) 1 (b) 2 (c) 4 (d) −1
17. Three numbers in AP have sum 18. The middle number is
    (a) 3 (b) 9 (c) 6 (d) 18
18. For an AP with $d=3$, $a_{10}-a_5$ is
    (a) 5 (b) 15 (c) 30 (d) 3
19. The number of two-digit multiples of 5 is
    (a) 17 (b) 18 (c) 19 (d) 20
20. The sum of 10 terms of an AP with first term 1 and last term 19 is
    (a) 100 (b) 200 (c) 190 (d) 95

Answer key

1 (b) · 2 (b) · 3 (c) · 4 (a) · 5 (b) · 6 (b) · 7 (c) · 8 (b) · 9 (b) · 10 (c) · 11 (a) · 12 (a) · 13 (b) · 14 (a) · 15 (b) · 16 (a) · 17 (c) · 18 (b) · 19 (b) · 20 (a)

Hints: Q3 $\sqrt{2},2\sqrt{2},3\sqrt{2},4\sqrt{2}$. Q4 $2x+20=5x+2$. Q8 $\frac{132}{4}+1$. Q11 $81-14\times 2$. Q15 $50-4(n-1)<0\Rightarrow n>13.5$. Q19 10, 15, …, 95.

Assertion–Reason (10)

(a) Both A and R are true and R is the correct explanation of A. (b) Both A and R are true but R is not the correct explanation of A. (c) A is true but R is false. (d) A is false but R is true.

1. A: 3, 6, 9, 12, … is an AP. R: The difference between consecutive terms is constant.
2. A: The 10th term of 5, 8, 11, … is 32. R: The nth term of an AP is $a+nd$.
3. A: 1, 4, 9, 16, … is an AP. R: All its terms are perfect squares.
4. A: If $S_n=3n^2+2n$, the second term is 11. R: $a_n=S_n-S_{n-1}$.
5. A: −150 is a term of 11, 8, 5, 2, … R: The position $n$ of a term must be a positive whole number.
6. A: The sum of the first 100 natural numbers is 5050. R: The sum of the first $n$ odd numbers is $n^2$.
7. A: If $a,b,c$ are in AP, then $b=\frac{a+c}{2}$. R: $a,b,c$ are in AP if and only if $2b=a+c$.
8. A: The common difference of an AP is always positive. R: For 10, 7, 4, …, $d=-3$.
9. A: 253 is a term of 3, 8, 13, … R: $253=3+50\times 5$.
10. A: The sum of the first $n$ terms of an AP is always positive. R: $S_n=\frac{n}{2}[2a+(n-1)d]$.

Key: 1 (a) · 2 (c) · 3 (d) · 4 (a) · 5 (d) · 6 (b) · 7 (a) · 8 (d) · 9 (a) · 10 (d)

Q2: $5+9\times 3=32$ ✓ but the formula is $a+(n-1)d$. Q4: $S_2-S_1=16-5=11$. Q5: $n-1=\frac{161}{3}$. Q10: −37, −33, … has negative sums.

Very short answer (1 mark each)

1. Find $d$ for $\sqrt{2},\sqrt{8},\sqrt{18},\dots$ Ans. $\sqrt{2}$
2. Find the 11th term of $-3,-\frac{1}{2},2,\dots$ Ans. 22
3. Write the next term of $a,3a,5a,\dots$ Ans. $7a$
4. Find the sum of the first 10 even natural numbers. Ans. $10\times 11=110$
5. If $k+2$, $4k-6$, $3k-2$ are in AP, find $k$. Ans. $8k-12=4k\Rightarrow k=3$
6. If $a_n=2n+3$, find $d$. Ans. 2
7. Which term of 5, 9, 13, … is 81? Ans. $\frac{76}{4}+1=$ 20th
8. Write the first four terms of the AP with $a=4$, $d=-3$. Ans. 4, 1, −2, −5

Short answer practice (2 marks each)

1. How many two-digit numbers are divisible by 3?

2. If 7 times the 7th term of an AP is equal to 11 times its 11th term, show that its 18th term is zero.

3. Find the sum $1+3+5+\cdots +199$.

Short answer practice (3 marks each)

4. Find the sum of all three-digit numbers divisible by 9.

5. The sum of the first $n$ terms of an AP is $3n^2+5n$. Find the AP and its 25th term.

6. Divide 24 into three parts in AP such that the product of the extremes is 55.

Long answer (5 marks each)

7. The sum of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last terms to the product of the two middle terms is 7 : 15. Find the numbers.

8. A man repays a loan of ₹3250 by paying ₹20 in the first month and then increasing the payment by ₹15 every month. How long will it take him to clear the loan?

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