If the sum of first m terms of an A.P. is same as sum of its first n terms (m ≠ n), then show that the sum of its first (m + n) terms is zero.

MathsCBSE Class 10 · Standard · Board exam 2024, set 30/2/1 · 3 marks · Short answer

CBSE Class 10 Mathematics Standard 2024: question 26(a)

Set in the CBSE Class 10 board exam of 2024, paper 30/2/1, section C, question 26(a), for 3 marks. Try it first; the hints and our step-by-step solution are below.

Q26(a) · 3 marks · Short answerArithmetic Progressions

If the sum of first $m$ terms of an A.P. is same as sum of its first $n$ terms $(m \neq = n)$ , then show that the sum of its first $(m + n)$ terms is zero.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 30/2/1.

Hints

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

Hint 1

Write both sums with $S_{k} = \frac{k}{2} [2 a + (k - 1) d]$ and equate.

Hint 2

Group terms so that $(m - n)$ is a common factor; you may cancel it since $m \neq = n$ .

Hint 3

Compare the result with the bracket in $S_{m + n}$ .

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.

Let first term $= a$ , common difference $= d$ . Given $S_{m} = S_{n}$ : $\frac{m}{2} [2 a + (m - 1) d] = \frac{n}{2} [2 a + (n - 1) d]$ .
1 mark
$\Rightarrow 2 a (m - n) + d [(m^{2} - n^{2}) - (m - n)] = 0 \Rightarrow 2 a (m - n) + d (m - n) (m + n - 1) = 0$ .
1 mark
Since $m \neq = n$ , divide by $(m - n)$ : $2 a + (m + n - 1) d = 0$ .
½ mark
$S_{m + n} = \frac{m + n}{2} [2 a + (m + n - 1) d] = \frac{m + n}{2} \times 0 = 0$ . Hence proved.
½ mark

Answer: $S_{m + n} = 0$ (shown)

Where the marks are

Writes $S_{m} = S_{n}$ using sum formula (1 mark)
Simplifies to $2 a (m - n) = - d (m - n) (m + n - 1)$ (1 mark)
Gets $2 a + (m + n - 1) d = 0$ (½ mark)
Concludes $S_{m + n} = 0$ (½ mark)

Revise the chapter

Chapter: Arithmetic Progressions (notes, formulas, where marks are lost and every past question)
Topic: Sum of First n Terms of an AP

Practise this chapterBoard-style questions from Arithmetic Progressions, one at a time, marked against the answer key and marking points.Back to the full paper2024 board paper, set 30/2/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2024: question 26(a)

Hints

Step-by-step solution

Revise the chapter

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