In an A.P., if S_n = 3n² + 5n and a_k = 164, find the value of k.

MathsCBSE Class 10 · Basic · Board exam 2023, set 430/4/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Basic 2023: question 33(B)

Set in the CBSE Class 10 board exam of 2023, paper 430/4/1, section D, question 33(B), for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q33(B) · 5 marks · Long answerArithmetic Progressions

In an A.P., if $S_{n} = 3 n^{2} + 5 n$ and $a_{k} = 164$ , find the value of $k$ .

Source: official question paper from cbse.gov.in (PDF). See it in the full 2023 board paper, set 430/4/1.

Hints

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

Hint 1

The sum of the first one term is the first term itself: $a_{1} = S_{1}$ .

Hint 2

Find $a_{2} = S_{2} - S_{1}$ , get d, then solve $a + (k - 1) d = 164$ .

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.

$S_{1} = 3 (1)^{2} + 5 (1) = 8$ , so $a_{1} = 8$
1 mark
$S_{2} = 3 (2)^{2} + 5 (2) = 22$ (sum of first two terms)
1 mark
$a_{1} + a_{2} = 22 \Rightarrow a_{2} = 22 - 8 = 14$
½ mark
$d = a_{2} - a_{1} = 14 - 8 = 6$
1 mark
$a_{k} = a + (k - 1) d = 164 \Rightarrow 8 + (k - 1) 6 = 164$
1 mark
$6 (k - 1) = 156 \Rightarrow k - 1 = 26 \Rightarrow k = 27$
½ mark

Answer: k = 27

Where the marks are

$S_{1} = a_{1} = 8$ (1 mark)
$S_{2} = 22$ (1 mark)
$a_{2} = 14$ (½ mark)
$d = 6$ (1 mark)
Forms $8 + (k - 1) 6 = 164$ (1 mark)
$k = 27$ (½ 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 430/4/1 (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 33(B)

Hints

Step-by-step solution

Revise the chapter

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