MathsCBSE Class 10 · Basic · Board exam 2023, set 430/2/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 35(a)

Set in the CBSE Class 10 board exam of 2023, paper 430/2/1, section D, question 35(a), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35(a) · 5 marks · Long answerArithmetic Progressions

If the sum of the first 7 terms of an A.P. is $- 14$ and that of 11 terms is $- 55$ , then find the sum of its first 'n' terms.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/2/1.

Hints

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

Hint 1

Write $S_{7}$ and $S_{11}$ using $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Hint 2

Simplify each to a linear equation in a and d and solve them together.

Hint 3

Substitute a and d back into the $S_{n}$ formula and simplify.

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.

Using $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ : $S_{7} = \frac{7}{2} (2 a + 6 d) = - 14 \Rightarrow 2 a + 6 d = - 4 \Rightarrow a + 3 d = - 2$ ...(i)
1 mark
$S_{11} = \frac{11}{2} (2 a + 10 d) = - 55 \Rightarrow 2 a + 10 d = - 10 \Rightarrow a + 5 d = - 5$ ...(ii)
1 mark
Subtracting (i) from (ii): $2 d = - 3 \Rightarrow d = - \frac{3}{2}$
1 mark
From (i): $a = - 2 - 3 (- \frac{3}{2}) = - 2 + \frac{9}{2} = \frac{5}{2}$
1 mark
$S_{n} = \frac{n}{2} [2 \cdot \frac{5}{2} + (n - 1) (- \frac{3}{2})] = \frac{n}{2} [\frac{10 - 3 n + 3}{2}] = \frac{n}{4} (13 - 3 n)$
1 mark

Answer: $S_{n} = \frac{n}{4} (13 - 3 n)$ (with $a = \frac{5}{2}$ , $d = - \frac{3}{2}$ )

Where the marks are

Forms equation for $S_{7}$ (1 mark)
Forms equation for $S_{11}$ (1 mark)
Finds $d = - \frac{3}{2}$ (1 mark)
Finds $a = \frac{5}{2}$ (1 mark)
Writes $S_{n} = \frac{n}{4} (13 - 3 n)$ (1 mark)

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 paper2023 board paper, set 430/2/1 (Basic): 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 Basic 2023: question 35(a)

Hints

Step-by-step solution

Revise the chapter

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