Arithmetic Progressions – Exercise 5.1

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This page provides comprehensive Arithmetic Progressions – Exercise 5.1. Free NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions Exercise 5-1. Step-by-step explained answers for CBSE Board exams. Download PDF and practice now.

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Question 1

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.

(ii) The amount of air present in a cylinder when a vacuum pump removes $\frac{1}{4}$ of the air remaining in the cylinder at a time.

(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.

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

(i) Fare for 1st km = 15. Fare for 2nd km = $15 + 8 = 23$. Fare for 3rd km = $23 + 8 = 31$.
List: $15, 23, 31, \dots$
Since difference is constant ($d=8$), it is an AP.

(ii) Let initial volume be $V$.
1st term: $V$.
2nd term: $V - \frac{1}{4}V = \frac{3}{4}V$.
3rd term: $\frac{3}{4}V - \frac{1}{4}(\frac{3}{4}V) = \frac{3}{4}V(1 - \frac{1}{4}) = (\frac{3}{4})^2 V$.
Common difference is not constant ($d_1 = -\frac{1}{4}V, d_2 = -\frac{3}{16}V$). Not an AP.

(iii) Cost: $150, 150+50, 150+50+50 \dots \Rightarrow 150, 200, 250 \dots$
Difference is constant ($d=50$). It is an AP.

(iv) Amount: $10000, 10000(1.08), 10000(1.08)^2 \dots$
This is a Geometric Progression (GP), not AP. Not an AP.

Questions 2 & 3

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

(i) $a = 10, d = 10$ (ii) $a = -2, d = 0$ (iii) $a = 4, d = -3$

(iv) $a = -1, d = \frac{1}{2}$ (v) $a = -1.25, d = -0.25$

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

(i) $3, 1, -1, -3, \dots$ (ii) $-5, -1, 3, 7, \dots$

(iii) $\frac{1}{3}, \frac{5}{3}, \frac{9}{3}, \frac{13}{3}, \dots$ (iv) $0.6, 1.7, 2.8, 3.9, \dots$

Question 2
(i) $10, 20, 30, 40$.
(ii) $-2, -2, -2, -2$.
(iii) $4, 1, -2, -5$.
(iv) $-1, -0.5, 0, 0.5$.
(v) $-1.25, -1.50, -1.75, -2.00$.

Question 3
(i) $a = 3, d = 1 - 3 = -2$.
(ii) $a = -5, d = -1 - (-5) = 4$.
(iii) $a = \frac{1}{3}, d = \frac{5}{3} - \frac{1}{3} = \frac{4}{3}$.
(iv) $a = 0.6, d = 1.7 - 0.6 = 1.1$.

Question 4

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

(i) $2, 4, 8, 16, \dots$

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

(iii) $-1.2, -3.2, -5.2, -7.2, \dots$

(iv) $-10, -6, -2, 2, \dots$

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

(vi) $0.2, 0.22, 0.222, 0.2222, \dots$

(vii) $0, -4, -8, -12, \dots$

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

(ix) $1, 3, 9, 27, \dots$

(x) $a, 2a, 3a, 4a, \dots$

(xi) $a, a^2, a^3, a^4, \dots$

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

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

(xiv) $1^2, 3^2, 5^2, 7^2, \dots$

(xv) $1^2, 5^2, 7^2, 73, \dots$

(i) $4-2=2, 8-4=4$. Not AP.
(ii) $d = \frac{5}{2} - 2 = 0.5$. Yes AP. Next: $4, 4.5, 5$.
(iii) $d = -2$. Yes AP. Next: $-9.2, -11.2, -13.2$.
(iv) $d = 4$. Yes AP. Next: $6, 10, 14$.
(v) $d = \sqrt{2}$. Yes AP. Next: $3+3\sqrt{2}, 3+4\sqrt{2}, 3+5\sqrt{2}$.
(vi) $0.22 - 0.2 = 0.02$; $0.222 - 0.22 = 0.002$. Not AP.
(vii) $d = -4$. Yes AP. Next: $-16, -20, -24$.
(viii) $d = 0$. Yes AP. Next: $-\frac{1}{2}, -\frac{1}{2}, -\frac{1}{2}$.
(ix) Ratio is 3 (GP). Not AP.
(x) $d = a$. Yes AP. Next: $5a, 6a, 7a$.
(xi) Powers of $a$. Not AP.
(xii) $\sqrt{2}, 2\sqrt{2}, 3\sqrt{2}, 4\sqrt{2}$. $d = \sqrt{2}$. Yes AP. Next: $5\sqrt{2}, 6\sqrt{2}, 7\sqrt{2}$ ($\sqrt{50}, \sqrt{72}, \sqrt{98}$).
(xiii) $\sqrt{3}, \sqrt{2}\sqrt{3}, 3, 2\sqrt{3}$. Not AP.
(xiv) $1, 9, 25, 49$. $9-1=8, 25-9=16$. Not AP.
(xv) $1, 25, 49, 73$. $25-1=24, 49-25=24$. $d=24$. Yes AP. Next: $97, 121, 145$.

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