Arithmetic Progressions | Exercise 5.1

Question 3

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

(i) 3,1,−1,−3,…3, 1, -1, -3, \dots

(ii) −5,−1,3,7,…-5, -1, 3, 7, \dots

(iii) 13,53,93,133,…\dfrac{1}{3}, \dfrac{5}{3}, \dfrac{9}{3}, \dfrac{13}{3}, \dots

(iv) 0.6,1.7,2.8,3.9,…0.6, 1.7, 2.8, 3.9, \dots

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Solution
Understand the Question
  • In an Arithmetic Progression (AP) with terms a1,a2,a3,…a_1, a_2, a_3, \dots:
    • First term (aa): The initial value in the sequence, a=a1a = a_1.
    • Common difference (dd): The difference between any consecutive terms, given by d=a2−a1=ak+1−akd = a_2 - a_1 = a_{k+1} - a_k.

(i) 3,1,−1,−3,…3, 1, -1, -3, \dots

Step 1 · Find First Term and Common Difference

Given AP: 3,1,−1,−3,…3, 1, -1, -3, \dots

First term: a=3a = 3

Common difference:

d=1−3=−2\begin{aligned} d &= 1 - 3 \\[0.6em] &= -2 \end{aligned}
Answer

(i) First term a=3a = 3, Common difference d=−2d = -2

(ii) −5,−1,3,7,…-5, -1, 3, 7, \dots

Step 1 · Find First Term and Common Difference

Given AP: −5,−1,3,7,…-5, -1, 3, 7, \dots

First term: a=−5a = -5

Common difference:

d=−1−(−5)=−1+5=4\begin{aligned} d &= -1 - (-5) \\[0.6em] &= -1 + 5 \\[0.6em] &= 4 \end{aligned}
Answer

(ii) First term a=−5a = -5, Common difference d=4d = 4

(iii) 13,53,93,133,…\dfrac{1}{3}, \dfrac{5}{3}, \dfrac{9}{3}, \dfrac{13}{3}, \dots

Step 1 · Find First Term and Common Difference

Given AP: 13,53,93,133,…\dfrac{1}{3}, \dfrac{5}{3}, \dfrac{9}{3}, \dfrac{13}{3}, \dots

First term: a=13a = \dfrac{1}{3}

Common difference:

d=53−13=43\begin{aligned} d &= \dfrac{5}{3} - \dfrac{1}{3} \\[0.6em] &= \dfrac{4}{3} \end{aligned}
Answer

(iii) First term a=13a = \dfrac{1}{3}, Common difference d=43d = \dfrac{4}{3}

(iv) 0.6,1.7,2.8,3.9,…0.6, 1.7, 2.8, 3.9, \dots

Step 1 · Find First Term and Common Difference

Given AP: 0.6,1.7,2.8,3.9,…0.6, 1.7, 2.8, 3.9, \dots

First term: a=0.6a = 0.6

Common difference:

d=1.7−0.6=1.1\begin{aligned} d &= 1.7 - 0.6 \\[0.6em] &= 1.1 \end{aligned}
Answer

(iv) First term a=0.6a = 0.6, Common difference d=1.1d = 1.1

Common Mistakes
  • Order of Subtraction: Incorrectly subtracting a1−a2a_1 - a_2 instead of a2−a1a_2 - a_1. For a decreasing sequence, the common difference dd must be negative.
  • Sign Error with Negatives: In part (ii), subtracting a negative number: −1−(−5)=−1+5=4-1 - (-5) = -1 + 5 = 4, rather than −1−5=−6-1 - 5 = -6.

More questions in Exercise 5.1

Q1

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 14\dfrac{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.

Q2
  1. Write first four terms of the AP, when the first term aa and the common difference dd are given as follows:

(i) a=10,d=10a = 10, \quad d = 10

(ii) a=−2,d=0a = -2, \quad d = 0

(iii) a=4,d=−3a = 4, \quad d = -3

(iv) a=−1,d=12a = -1, \quad d = \dfrac{1}{2}

(v) a=−1.25,d=−0.25a = -1.25, \quad d = -0.25

Q3
  1. For the following APs, write the first term and the common difference:

(i) 3,1,−1,−3,…3, 1, -1, -3, \dots

(ii) −5,−1,3,7,…-5, -1, 3, 7, \dots

(iii) 13,53,93,133,…\dfrac{1}{3}, \dfrac{5}{3}, \dfrac{9}{3}, \dfrac{13}{3}, \dots

(iv) 0.6,1.7,2.8,3.9,…0.6, 1.7, 2.8, 3.9, \dots

Q4
  1. Which of the following are APs ? If they form an AP, find the common difference dd and write three more terms.

(i) 2,4,8,16,…2, 4, 8, 16, \dots

(ii) 2,52,3,72,…2, \dfrac{5}{2}, 3, \dfrac{7}{2}, \dots

(iii) −1.2,−3.2,−5.2,−7.2,…-1.2, -3.2, -5.2, -7.2, \dots

(iv) −10,−6,−2,2,…-10, -6, -2, 2, \dots

(v) 3,3+2,3+22,3+32,…3, 3 + \sqrt{2}, 3 + 2\sqrt{2}, 3 + 3\sqrt{2}, \dots

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

(vii) 0,−4,−8,−12,…0, -4, -8, -12, \dots

(viii) −12,−12,−12,−12,…-\dfrac{1}{2}, -\dfrac{1}{2}, -\dfrac{1}{2}, -\dfrac{1}{2}, \dots

(ix) 1,3,9,27,…1, 3, 9, 27, \dots

(x) a,2a,3a,4a,…a, 2a, 3a, 4a, \dots

(xi) a,a2,a3,a4,…a, a^2, a^3, a^4, \dots

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

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

(xiv) 12,32,52,72,…1^2, 3^2, 5^2, 7^2, \dots

(xv) 12,52,72,73,…1^2, 5^2, 7^2, 73, \dots

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