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NCERT Solutions · Class 10 · Chapter 5: Arithmetic Progressions

NCERT Solutions for Class 10 Maths Chapter 5 Exercise 5.4

Exercise 5.4: Optional: harder problems. Harder uses of \(a_n\) and \(S_n\): inequalities for "first negative term", two conditions on two unknowns, and turning a physical set-up into an AP.

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Exercise 5.4 questions and solutions

Exercise 5.4, Question 1

Which term of \(121, 117, 113, \ldots\) is its first negative term?
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  1. \(a_n = 121 - 4(n-1) = 125 - 4n\).
  2. \(125 - 4n < 0 \Rightarrow n > 31.25\), so the first such \(n\) is \(32\) (\(a_{32} = -3\)).
Answer: The 32nd term

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Exercise 5.4, Question 2

The 3rd and 7th terms of an AP add to \(6\) and multiply to \(8\). Find the sum of the first 16 terms.
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  1. \(2a + 8d = 6 \Rightarrow a + 4d = 3\), so the terms are \(3 - 2d\) and \(3 + 2d\).
  2. \[\begin{aligned}&(3 - 2d)(3 + 2d) = 8 \\ \Rightarrow\ &9 - 4d^2 = 8 \\ \Rightarrow\ &d = \pm\tfrac12\end{aligned}\]
  3. \(d = \tfrac12\): \(a = 1\), \(S_{16} = 8(2 + 7.5) = 76\).
  4. \(d = -\tfrac12\): \(a = 5\), \(S_{16} = 8(10 - 7.5) = 20\).
Answer: \(76\) or \(20\)

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Exercise 5.4, Question 3

Ladder rungs are 25 cm apart and shrink evenly from 45 cm at the bottom to 25 cm at the top; top and bottom rungs are \(2\tfrac12\) m apart. How much wood do the rungs need?
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  1. \(250 \div 25 = 10\) gaps, so \(11\) rungs.
  2. Total length \(= \tfrac{11}{2}(45 + 25) = 385\) cm.
Answer: \(385\) cm (3.85 m)

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Exercise 5.4, Question 4

Houses in a row are numbered 1 to 49. Show there is a house \(x\) with the sum of the numbers before it equal to the sum of those after it, and find \(x\).
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  1. Before: \(1 + \cdots + (x-1) = \tfrac{x(x-1)}{2}\). After: \[\tfrac{49 \times 50}{2} - \tfrac{x(x+1)}{2}\]
  2. Set equal: \[\begin{aligned}&x^2 - x = 2450 - x^2 - x \\ \Rightarrow\ &x^2 = 1225\end{aligned}\]
  3. \(x = 35\), a house number between 1 and 49.
Answer: \(x = 35\)

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Exercise 5.4, Question 5

A terrace has 15 concrete steps, each 50 m long, with a rise of \(\tfrac14\) m and a tread of \(\tfrac12\) m. Find the volume of concrete.
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  1. Step \(k\) is a block \(50 \times \tfrac12 \times \tfrac k4\), volume \(6.25k\ \text{m}^3\).
  2. Total \[\begin{aligned}= 6.25(1 + 2 + \cdots + 15) &= 6.25 \times 120 \\ &= 750\end{aligned}\]
Answer: \(750\ \text{m}^3\)

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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

Textbook: NCERT Mathematics Class 10 (rationalised edition, 2023-24 reprint onward), free from ncert.nic.in. Question statements are shortened to the minimum needed; the solutions and tips are our own. CBSE Math Revision is independent and not affiliated with NCERT or CBSE. Spotted a slip? Tell us and it goes in the corrections log.