NCERT Class 10 Maths – Exercise 5.1 Solutions

NCERT Class 10 Maths

Chapter 5 – Arithmetic Progressions | Exercise 5.1

(Rationalized Syllabus 2025-26)

In which of the following situations, does the list of numbers involved make an arithmetic progression, and why?

(i) The taxi fare after each km when the fare is ₹15 for the first km and ₹8 for each additional km.

Let $a_n$ be the fare for $n$ km.

$$a_1 = 15$$

$$a_2 = 15 + 8 = 23$$

$$a_3 = 23 + 8 = 31$$

Series: 15, 23, 31, …

Since the difference between consecutive terms is constant ($d=8$), it is an AP.

✔️ Yes, it is an AP

(ii) The amount of air present in a cylinder when a vacuum pump removes 1/4 of the air remaining in the cylinder at a time.

Let initial volume be $V$.

$$a_1 = V$$

$$a_2 = V – \frac{1}{4}V = \frac{3}{4}V$$

$$a_3 = \frac{3}{4}V – \frac{1}{4}(\frac{3}{4}V) = \frac{3}{4}V(1 – \frac{1}{4}) = (\frac{3}{4})^2V$$

Ratio is constant ($\frac{3}{4}$), but the difference is not constant.

✖️ No, it is not an AP (It’s a GP)

(iii) The cost of digging a well after every metre of digging, when it costs ₹150 for the first metre and rises by ₹50 for each subsequent metre.

Cost for 1st meter = 150

Cost for 2nd meter = 150 + 50 = 200

Cost for 3rd meter = 200 + 50 = 250

Series: 150, 200, 250, … (Common difference $d=50$).

✔️ Yes, it is an AP

(iv) The amount of money in the account every year, when ₹10000 is deposited at compound interest at 8% per annum.

Amount after 1 year: $10000(1 + \frac{8}{100})$

Amount after 2 years: $10000(1 + \frac{8}{100})^2$

The difference between terms increases every year.

✖️ No, it is not an AP

Write first four terms of the AP, when the first term $a$ and the common difference $d$ are given as follows:

(i) $a = 10, d = 10$

$$a_1 = 10$$

$$a_2 = 10 + 10 = 20$$

$$a_3 = 20 + 10 = 30$$

$$a_4 = 30 + 10 = 40$$

✔️ 10, 20, 30, 40

(ii) $a = -2, d = 0$

$$a_1 = -2$$

$$a_2 = -2 + 0 = -2$$

$$a_3 = -2 + 0 = -2$$

$$a_4 = -2 + 0 = -2$$

✔️ -2, -2, -2, -2

(iii) $a = 4, d = -3$

$$a_1 = 4$$

$$a_2 = 4 + (-3) = 1$$

$$a_3 = 1 + (-3) = -2$$

$$a_4 = -2 + (-3) = -5$$

✔️ 4, 1, -2, -5

(iv) $a = -1, d = \frac{1}{2}$

$$a_1 = -1$$

$$a_2 = -1 + 0.5 = -0.5$$

$$a_3 = -0.5 + 0.5 = 0$$

$$a_4 = 0 + 0.5 = 0.5$$

✔️ -1, -0.5, 0, 0.5

(v) $a = -1.25, d = -0.25$

$$a_1 = -1.25$$

$$a_2 = -1.25 – 0.25 = -1.50$$

$$a_3 = -1.50 – 0.25 = -1.75$$

$$a_4 = -1.75 – 0.25 = -2.00$$

✔️ -1.25, -1.50, -1.75, -2.00

For the following APs, write the first term and the common difference:

(i) $3, 1, -1, -3, …$

First term $a = 3$.

Common difference $d = a_2 – a_1 = 1 – 3 = -2$.

✔️ a = 3, d = -2

(ii) $-5, -1, 3, 7, …$

First term $a = -5$.

Common difference $d = -1 – (-5) = -1 + 5 = 4$.

✔️ a = -5, d = 4

(iii) $\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, …$

First term $a = \frac{1}{3}$.

Common difference $d = \frac{5}{3} – \frac{1}{3} = \frac{4}{3}$.

✔️ a = 1/3, d = 4/3

(iv) $0.6, 1.7, 2.8, 3.9, …$

First term $a = 0.6$.

Common difference $d = 1.7 – 0.6 = 1.1$.

✔️ a = 0.6, d = 1.1

Which of the following are APs? If they form an AP, find the common difference $d$ and write three more terms.

(i) $2, 4, 8, 16, …$

$4-2=2$, $8-4=4$. Difference is not constant.

✖️ Not an AP

(ii) $2, \frac{5}{2}, 3, \frac{7}{2}, …$

$d = \frac{5}{2} – 2 = 0.5$. $3 – \frac{5}{2} = 0.5$. It is an AP.

Next terms: $3.5 + 0.5 = 4$, then $4.5 (\frac{9}{2})$, then $5$.

✔️ Yes, d = 1/2. Next terms: 4, 9/2, 5

(iii) $-1.2, -3.2, -5.2, -7.2, …$

$d = -3.2 – (-1.2) = -2$. Constant difference.

Next terms: $-9.2, -11.2, -13.2$.

✔️ Yes, d = -2. Next terms: -9.2, -11.2, -13.2

(iv) $-10, -6, -2, 2, …$

$d = -6 – (-10) = 4$. Constant difference.

Next terms: $2+4=6$, $6+4=10$, $10+4=14$.

✔️ Yes, d = 4. Next terms: 6, 10, 14

(v) $3, 3+\sqrt{2}, 3+2\sqrt{2}, 3+3\sqrt{2}, …$

$d = (3+\sqrt{2}) – 3 = \sqrt{2}$. Constant difference.

Next terms: $3+4\sqrt{2}, 3+5\sqrt{2}, 3+6\sqrt{2}$.

✔️ Yes, d = $\sqrt{2}$. Next terms: $3+4\sqrt{2}, 3+5\sqrt{2}, 3+6\sqrt{2}$

(vi) $0.2, 0.22, 0.222, 0.2222, …$

$0.22 – 0.2 = 0.02$. $0.222 – 0.22 = 0.002$. Difference not constant.

✖️ Not an AP

(vii) $0, -4, -8, -12, …$

$d = -4 – 0 = -4$. Constant difference.

Next terms: $-16, -20, -24$.

✔️ Yes, d = -4. Next terms: -16, -20, -24

(viii) $-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}, …$

$d = 0$. Constant difference.

✔️ Yes, d = 0. Next terms: -1/2, -1/2, -1/2

(ix) $1, 3, 9, 27, …$

$3-1=2$, $9-3=6$. Difference not constant.

✖️ Not an AP

(x) $a, 2a, 3a, 4a, …$

$d = 2a – a = a$. Constant difference.

Next terms: $5a, 6a, 7a$.

✔️ Yes, d = a. Next terms: 5a, 6a, 7a

(xi) $a, a^2, a^3, a^4, …$

$a^2 – a \neq a^3 – a^2$. Difference not constant.

✖️ Not an AP

(xii) $\sqrt{2}, \sqrt{8}, \sqrt{18}, \sqrt{32}, …$

Simplify terms: $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}, …$

$d = 2\sqrt{2} – \sqrt{2} = \sqrt{2}$. Constant difference.

Next terms: $5\sqrt{2} (\sqrt{50}), 6\sqrt{2} (\sqrt{72}), 7\sqrt{2} (\sqrt{98})$.

✔️ Yes, d = $\sqrt{2}$. Next terms: $\sqrt{50}, \sqrt{72}, \sqrt{98}$

(xiii) $\sqrt{3}, \sqrt{6}, \sqrt{9}, \sqrt{12}, …$

$\sqrt{6}-\sqrt{3} \neq 3-\sqrt{6}$. Difference not constant.

✖️ Not an AP

(xiv) $1^2, 3^2, 5^2, 7^2, …$

Values: $1, 9, 25, 49$. Differences: $8, 16, 24$. Not constant.

✖️ Not an AP

(xv) $1^2, 5^2, 7^2, 73, …$

Values: $1, 25, 49, 73$. Differences: $24, 24, 24$. Constant.

Next terms: $73+24=97$, $97+24=121$, $121+24=145$.

✔️ Yes, d = 24. Next terms: 97, 121, 145

🎉 Exercise 5.1 Completed