The sum of first n terms of an AP is 5n² - n. Find the n^th term of the AP.

MathsCBSE Class 10 · Basic · Sample paper, session 2025-26 · 3 marks · Short answer

CBSE Class 10 Mathematics Basic sample paper 2025-26: question 31

From the official CBSE sample question paper for 2025-26, section C, question 31, for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q31 · 3 marks · Short answerArithmetic Progressions

The sum of first $n$ terms of an AP is $5 n^{2} - n$ . Find the $n^{t h}$ term of the AP.

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2025-26.

Hints

Open one hint at a time and try again before the next.

Hint 1

The $n$ th term is the sum of $n$ terms minus the sum of $(n - 1)$ terms.

Hint 2

Replace $n$ by $(n - 1)$ in $5 n^{2} - n$ and subtract carefully.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

$n$ th term: $a_{n} = S_{n} - S_{n - 1}$
½ mark
$a_{n} = [5 n^{2} - n] - [5 (n - 1)^{2} - (n - 1)] = 5 [n^{2} - (n - 1)^{2}] - [n - (n - 1)]$
1 mark
$= 5 (2 n - 1) - 1 = 10 n - 6$ . So $a_{n} = 10 n - 6$ . (Check: $a_{1} = S_{1} = 4$ .)
1½ marks

Answer: $a_{n} = 10 n - 6$

Where the marks are

Uses $a_{n} = S_{n} - S_{n - 1}$ (½ mark)
Substitutes $S_{n - 1}$ correctly (1 mark)
Simplifies to $a_{n} = 10 n - 6$ (1½ marks)

Revise the chapter

Chapter: Arithmetic Progressions (notes, formulas, where marks are lost and every past question)
Topic: Finding nth Term from the Sum of n Terms

Practise this chapterBoard-style questions from Arithmetic Progressions, one at a time, marked against the answer key and marking points.Back to the full paperSample paper 2025-26 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic sample paper 2025-26: question 31

Hints

Step-by-step solution

Revise the chapter

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