Arithmetic Progressions Class 10 Exercise 5.4 Solutions

The NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions Exercise 5.4 work out all 5 optional questions that apply the AP and sum formulas to word problems on salaries, ladders, house numbering, and concrete terraces. Every answer follows the 2026-27 CBSE syllabus and shows the full working the board expects.

• Questions covered: 5 in total, including finding the first negative term, working backwards from a sum-and-product condition, a ladder wood-length problem, a house-number balance proof, and a concrete-volume AP problem.
• Core skills: converting word problems into AP equations, applying S_n = (n/2)(a + l) and a_n = a + (n − 1)d, and working with both positive and negative common differences.
• Board value: Exercise 5.4 is optional but its question types (first negative term, sum-product conditions, word problems) appear regularly as 3-mark and 4-mark questions in CBSE Class 10 board papers.

What Exercise 5.4 of Arithmetic Progressions Covers for Class 10

Exercise 5.4 is the optional set of the chapter, but the question types it uses are not optional at all in the board paper. These 5 questions test whether students can move beyond drilling formulas into actually using them in realistic situations. Each question wraps the AP idea inside a story, so the first job is always to identify a, d, and n before touching S_n or a_n.

• Q1: find the first negative term of the AP 121, 117, 113, ... using the inequality a_n < 0.
• Q2: recover a and d from a sum-and-product condition on the 3rd and 7th terms, then compute S₁₆.
• Q3: ladder rung wood problem, counting rungs from spacing then summing an AP of rung lengths.
• Q4: house numbering balance problem, showing a balanced house exists and finding its number using S_x-1 = S₄₉ − S_x.
• Q5: concrete terrace volume problem, where each step's volume forms an AP with the same common difference as the first step's volume.

How to Solve Exercise 5.4 Word Problems Question by Question

Every Exercise 5.4 question follows the same pattern: extract a, d, and n (or l) from the problem's story, then pick the right AP formula. The two formulas you need here are the nth term a_n = a + (n − 1)d and the sum S_n = (n/2)(2a + (n − 1)d) or the shortcut S_n = (n/2)(a + l) when both ends are known.

First Negative Term and Inequality Method in Arithmetic Progressions Exercise 5.4

Question 1 introduces a technique that trips up many students: finding when an AP first crosses a boundary. The trick is to write the term formula a_n = 125 − 4n as an inequality rather than an equation. Setting 125 − 4n < 0 gives n > 31.25. Since n must be a whole number, the answer is n = 32, not n = 31.25. Always round up when the boundary falls between integers.

• Set up the inequality: a_n = a + (n − 1)d < 0, then solve for n.
• Round up, not down: n = 31.25 means a₃₁ = 1 is still positive, so the 32nd term is the first to go negative.

Sum-and-Product Conditions and Both Solutions for Arithmetic Progressions Exercise 5.4 Q2

Question 2 is the most algebraically demanding in this exercise. The key is to substitute a = 3 − 4d into the product condition, which turns the product into a difference of squares (3 − 2d)(3 + 2d) = 9 − 4d². Setting this equal to 8 gives 4d² = 1, so d = +1/2 or d = −1/2. Both are valid, and both give a different S₁₆, so students must report both answers.

• Case 1: d = 1/2, a = 1. Then S₁₆ = 8[2 + 7.5] = 8(9.5) = 76.
• Case 2: d = −1/2, a = 5. Then S₁₆ = 8[10 − 7.5] = 8(2.5) = 20.

Arithmetic Progressions Exercise 5.4 Marks and Board Exam Trends for Class 10

Exercise 5.4 is optional, but the CBSE board paper draws from these question types often.

Common Mistakes Students Make in Arithmetic Progressions Exercise 5.4

Most marks lost in Exercise 5.4 come from a few habits worth fixing before the exam.

• Rounding down in Q1: n > 31.25 means n = 32, not 31. The word "first" in "first negative term" means take the next whole number above the fractional boundary, not the one below it.
• Reporting only one S₁₆ in Q2: d² = 1/4 gives two values of d. Both are genuine and both give a different sum. Writing only 76 loses half the marks for Q2.
• Gaps vs. rungs in Q3: 250 ÷ 25 = 10 gaps, but there are 11 rungs. The first and last rung are at the two ends, so rungs = gaps + 1. This mistake alone changes the answer by one rung length.
• Unit mismatch in Q3: mixing metres and centimetres when computing rung lengths gives a wrong sum. Convert 2½ m = 250 cm first.
• Missing the −1 in Q4: the sum of houses before house x is S_x−1, not S_x. Using S_x on the left side gives a different equation with no clean solution.

Other Resources for This Chapter Class 10 Maths Arithmetic Progressions

Pair Exercise 5.4 with the other Class 10 Maths resources for this chapter, all linked below.

NCERT Solutions for Class 10 Maths Arithmetic Progressions: All Exercises

The chapter has four exercises. The table below links each to its own step-by-step solutions page.

NCERT Solutions for Class 10 Maths: All Chapters

Once Arithmetic Progressions is done, move on to the other chapters. Each link opens that chapter's NCERT Solutions page.

All NCERT Solutions for Class 10 Maths Chapter 5 Arithmetic Progressions Exercise 5.4 with Step-by-Step Solutions

Questions