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NCERT Solutions · Class 10 · Chapter 5: Arithmetic Progressions

NCERT Solutions for Class 10 Maths Chapter 5 Exercise 5.2

Exercise 5.2: The nth term. \(a_n = a + (n-1)d\). Two facts about an AP give two equations in \(a\) and \(d\); "which term" questions solve for \(n\), which must be a positive whole number.

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Exercise 5.2 questions and solutions

Exercise 5.2, Question 1

Fill in the blanks, where \(a\) is the first term, \(d\) the common difference and \(a_n\) the nth term.
(i) \(a = 7,\ d = 3,\ n = 8,\ a_n = ?\)
Show solution
  1. \(a_8 = 7 + 7(3) = 28\).
Answer: \(a_n = 28\)
(ii) \(a = -18,\ n = 10,\ a_n = 0,\ d = ?\)
Show solution
  1. \(0 = -18 + 9d \Rightarrow d = 2\).
Answer: \(d = 2\)
(iii) \(d = -3,\ n = 18,\ a_n = -5,\ a = ?\)
Show solution
  1. \[\begin{aligned}&-5 = a + 17(-3) \\ \Rightarrow\ &a = -5 + 51 = 46\end{aligned}\]
Answer: \(a = 46\)
(iv) \(a = -18.9,\ d = 2.5,\ a_n = 3.6,\ n = ?\)
Show solution
  1. \(3.6 = -18.9 + (n-1)(2.5)\).
  2. \[\begin{aligned}&(n-1)(2.5) = 22.5 \\ \Rightarrow\ &n - 1 = 9 \\ \Rightarrow\ &n = 10\end{aligned}\]
Answer: \(n = 10\)
(v) \(a = 3.5,\ d = 0,\ n = 105,\ a_n = ?\)
Show solution
  1. With \(d = 0\) every term is \(3.5\).
Answer: \(a_n = 3.5\)

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Exercise 5.2, Question 2

Choose the correct answer.
(i) The 30th term of \(10, 7, 4, \ldots\) is: (A) 97 (B) 77 (C) −77 (D) −87
Show solution
  1. \(a = 10,\ d = -3\).
  2. \(a_{30} = 10 + 29(-3) = -77\).
Answer: (C) \(-77\)
(ii) The 11th term of \(-3, -\tfrac12, 2, \ldots\) is: (A) 28 (B) 22 (C) −38 (D) \(-48\tfrac12\)
Show solution
  1. \(a = -3,\ d = \tfrac52\).
  2. \(a_{11} = -3 + 10 \times \tfrac52 = 22\).
Answer: (B) \(22\)

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Exercise 5.2, Question 3

Find the missing terms (shown as □) of each AP.
(i) \(2,\ □,\ 26\)
Show solution
  1. \(a_3 = a + 2d\): \(26 = 2 + 2d \Rightarrow d = 12\).
  2. Middle term \(= 2 + 12 = 14\).
Answer: \(14\)
(ii) \(□,\ 13,\ □,\ 3\)
Show solution
  1. \(a_2 = a + d = 13\) and \(a_4 = a + 3d = 3\).
  2. Subtract: \(2d = -10 \Rightarrow d = -5\), so \(a = 18\).
  3. \(a_3 = 13 - 5 = 8\).
Answer: \(18\) and \(8\)
(iii) \(5,\ □,\ □,\ 9\tfrac12\)
Show solution
  1. \(a_4 = 5 + 3d = 9.5 \Rightarrow d = 1.5\).
  2. Terms: \(6.5\) and \(8\).
Answer: \(6\tfrac12\) and \(8\)
(iv) \(-4,\ □,\ □,\ □,\ □,\ 6\)
Show solution
  1. \(a_6 = -4 + 5d = 6 \Rightarrow d = 2\).
  2. Terms: \(-2, 0, 2, 4\).
Answer: \(-2, 0, 2, 4\)
(v) \(□,\ 38,\ □,\ □,\ □,\ -22\)
Show solution
  1. \(a + d = 38\) and \(a + 5d = -22\).
  2. Subtract: \(4d = -60 \Rightarrow d = -15\), so \(a = 53\).
  3. Terms: \(53, 38, 23, 8, -7, -22\).
Answer: \(53, 23, 8, -7\)

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Exercise 5.2, Question 4

Which term of \(3, 8, 13, 18, \ldots\) is \(78\)?
Show solution
  1. \(a = 3,\ d = 5\).
  2. \[\begin{aligned}&3 + (n-1)5 = 78 \\ \Rightarrow\ &n - 1 = 15 \\ \Rightarrow\ &n = 16\end{aligned}\]
Answer: The 16th term

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Exercise 5.2, Question 5

Find the number of terms in each AP.
(i) \(7, 13, 19, \ldots, 205\)
Show solution
  1. \(a = 7,\ d = 6,\ a_n = 205\).
  2. \[\begin{aligned}&7 + (n-1)6 = 205 \\ \Rightarrow\ &n - 1 = 33 \\ \Rightarrow\ &n = 34\end{aligned}\]
Answer: \(34\) terms
(ii) \(18, 15\tfrac12, 13, \ldots, -47\)
Show solution
  1. \(a = 18,\ d = -\tfrac52\).
  2. \[\begin{aligned}&18 - \tfrac52(n-1) = -47 \\ \Rightarrow\ &\tfrac52(n-1) = 65 \\ \Rightarrow\ &n - 1 = 26\end{aligned}\]
Answer: \(27\) terms

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Exercise 5.2, Question 6

Is \(-150\) a term of \(11, 8, 5, 2, \ldots\)?
Show solution
  1. \(a = 11,\ d = -3\).
  2. \[\begin{aligned}&11 - 3(n-1) = -150 \\ \Rightarrow\ &3(n-1) = 161 \\ \Rightarrow\ &n = \tfrac{164}{3}\end{aligned}\]
  3. \(n\) is not a whole number.
Answer: No, \(-150\) is not a term.

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Exercise 5.2, Question 7

The 11th term of an AP is \(38\) and the 16th term is \(73\). Find the 31st term.
Show solution
  1. \(a + 10d = 38\) and \(a + 15d = 73\).
  2. Subtract: \(5d = 35 \Rightarrow d = 7\), so \(a = -32\).
  3. \(a_{31} = -32 + 30(7) = 178\).
Answer: \(178\)

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Exercise 5.2, Question 8

An AP has 50 terms; its 3rd term is \(12\) and its last term is \(106\). Find the 29th term.
Show solution
  1. \(a + 2d = 12\) and \(a + 49d = 106\).
  2. Subtract: \(47d = 94 \Rightarrow d = 2\), so \(a = 8\).
  3. \(a_{29} = 8 + 28(2) = 64\).
Answer: \(64\)

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Exercise 5.2, Question 9

The 3rd and 9th terms of an AP are \(4\) and \(-8\). Which term is zero?
Show solution
  1. \(a + 2d = 4\) and \(a + 8d = -8\).
  2. Subtract: \(6d = -12 \Rightarrow d = -2\), so \(a = 8\).
  3. \(8 - 2(n-1) = 0 \Rightarrow n = 5\).
Answer: The 5th term

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Exercise 5.2, Question 10

The 17th term of an AP is \(7\) more than its 10th term. Find \(d\).
Show solution
  1. \[\begin{aligned}a_{17} - a_{10} &= (a + 16d) - (a + 9d) \\ &= 7d\end{aligned}\]
  2. \(7d = 7 \Rightarrow d = 1\).
Answer: \(d = 1\)

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Exercise 5.2, Question 11

Which term of \(3, 15, 27, 39, \ldots\) is \(132\) more than its 54th term?
Show solution
  1. \(d = 12\).
  2. \[\begin{aligned}&a_n - a_{54} = (n - 54)d = 132 \\ \Rightarrow\ &n - 54 = 11\end{aligned}\]
Answer: The 65th term

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Exercise 5.2, Question 12

Two APs have the same common difference. Their 100th terms differ by \(100\). How far apart are their 1000th terms?
Show solution
  1. Let the first terms be \(a\) and \(b\), common difference \(d\).
  2. 100th terms: \((a + 99d) - (b + 99d) = a - b = 100\).
  3. 1000th terms: \((a + 999d) - (b + 999d) = a - b\).
Answer: \(100\): the difference between corresponding terms never changes.

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Exercise 5.2, Question 13

How many three-digit numbers are divisible by \(7\)?
Show solution
  1. The smallest is \(105\) and the largest is \(994\): \(105, 112, \ldots, 994\) is an AP with \(d = 7\).
  2. \[\begin{aligned}&994 = 105 + (n-1)7 \\ \Rightarrow\ &n - 1 = 127\end{aligned}\]
Answer: \(128\)

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Exercise 5.2, Question 14

How many multiples of \(4\) lie between \(10\) and \(250\)?
Show solution
  1. The first is \(12\) and the last is \(248\): an AP with \(d = 4\).
  2. \(248 = 12 + (n-1)4 \Rightarrow n - 1 = 59\).
Answer: \(60\)

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Exercise 5.2, Question 15

For which \(n\) are the nth terms of \(63, 65, 67, \ldots\) and \(3, 10, 17, \ldots\) equal?
Show solution
  1. \(63 + 2(n-1) = 3 + 7(n-1)\).
  2. \(60 = 5(n-1) \Rightarrow n = 13\).
Answer: \(n = 13\)

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Exercise 5.2, Question 16

Find the AP whose 3rd term is \(16\) and whose 7th term is \(12\) more than its 5th term.
Show solution
  1. \(a_7 - a_5 = 2d = 12 \Rightarrow d = 6\).
  2. \(a + 2(6) = 16 \Rightarrow a = 4\).
Answer: \(4, 10, 16, 22, \ldots\)

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Exercise 5.2, Question 17

Find the 20th term from the last term of \(3, 8, 13, \ldots, 253\).
Show solution
  1. Read the AP backwards: \(253, 248, 243, \ldots\), so \(a = 253,\ d = -5\).
  2. 20th term \(= 253 + 19(-5) = 158\).
Answer: \(158\)

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Exercise 5.2, Question 18

The 4th and 8th terms of an AP add to \(24\); the 6th and 10th terms add to \(44\). Find the first three terms.
Show solution
  1. \(2a + 10d = 24\) and \(2a + 14d = 44\).
  2. Subtract: \(4d = 20 \Rightarrow d = 5\), so \(a = -13\).
Answer: \(-13, -8, -3\)

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Exercise 5.2, Question 19

A salary starts at ₹5000 a year in 1995 and rises by ₹200 each year. In which year does it reach ₹7000?
Show solution
  1. Salaries form an AP: \(a = 5000,\ d = 200\).
  2. \[\begin{aligned}&5000 + 200(n-1) = 7000 \\ \Rightarrow\ &n = 11\end{aligned}\]
  3. The 11th year counting 1995 as the first is 2005.
Answer: In 2005

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Exercise 5.2, Question 20

Savings are ₹5 in week 1 and rise by ₹1.75 each week. In which week are they ₹20.75?
Show solution
  1. \[\begin{aligned}&5 + 1.75(n-1) = 20.75 \\ \Rightarrow\ &1.75(n-1) = 15.75\end{aligned}\]
  2. \(n - 1 = 9\).
Answer: Week \(10\)

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Done the NCERT exercises? The board paper asks more

Arithmetic Progressions has 39 original board-style questions (MCQ, assertion–reason, short and long answers, case studies) with step mark schemes, revision notes and a four-step Route to 95. Three sample questions are open to everyone; a free account opens the rest.

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Arithmetic Progressions in our sample papers: Sample paper 1 (questions 5, 20, 32, 36) · Sample paper 2 (questions 5, 6, 32) · Sample paper 3 (questions 6, 7, 32, 38) · Sample paper 4 (questions 6, 7, 27, 32) · Sample paper 5 (questions 6, 32).

Also useful: free MCQs and case studies for Arithmetic Progressions · Class 10 formula sheet · official CBSE board and sample papers · our sample papers with marking scheme · the Route to 95 plan

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