How many terms of the A.P. 27, 24, 21, … must be taken so that their sum is 105? Which term of the A.P. is zero?

MathsCBSE Class 10 · Basic · Board exam 2024, set 430/1/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2024: question 33

Set in the CBSE Class 10 board exam of 2024, paper 430/1/1, section D, question 33, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33 · 5 marks · Long answerArithmetic Progressions

How many terms of the A.P. $27, 24, 21, \dots$ must be taken so that their sum is $105$ ? Which term of the A.P. is zero?

Source: official question paper from cbse.gov.in (PDF). See it in the full 2024 board paper, set 430/1/1.

Hints

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

Hint 1

Find the common difference and use $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ .

Hint 2

You will get a quadratic in n; both roots may be acceptable — check why.

Hint 3

For the zero term, set $a_{n} = a + (n - 1) d = 0$ .

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.

Here $a = 27$ , $d = 24 - 27 = - 3$ . Let n terms be required.
½ mark
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] \Rightarrow 105 = \frac{n}{2} [54 + (n - 1) (- 3)] = \frac{n}{2} (57 - 3 n)$
1½ marks
$210 = 57 n - 3 n^{2} \Rightarrow 3 n^{2} - 57 n + 210 = 0 \Rightarrow n^{2} - 19 n + 70 = 0$
1 mark
$(n - 5) (n - 14) = 0 \Rightarrow n = 5$ or $n = 14$ . Both are valid: the terms from the 6th to the 14th $(12, 9, \dots, - 12)$ add up to zero, so 5 terms or 14 terms give sum 105.
1 mark
$a_{n} = 0 \Rightarrow 27 + (n - 1) (- 3) = 0 \Rightarrow n - 1 = 9 \Rightarrow n = 10$ . The 10th term is zero.
1 mark

Answer: 5 terms or 14 terms; the 10th term is zero

Where the marks are

d = −3 (½ mark)
Sets up $105 = \frac{n}{2} [54 + (n - 1) (- 3)]$ (1½ marks)
Simplifies to $n^{2} - 19 n + 70 = 0$ (1 mark)
n = 5 or 14 (1 mark)
$a_{10} = 0$ (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 paper2024 board paper, set 430/1/1 (Basic): every question, by section.

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CBSE Class 10 Mathematics Basic 2024: question 33

Hints

Step-by-step solution

Revise the chapter

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