In an A.P., the sum of the first 'n' terms is 3n² + n. Find the first term and the common difference of the A.P. Hence, find its 15th term.

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

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

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

Q35(b) · 5 marks · Long answerArithmetic Progressions

In an A.P., the sum of the first 'n' terms is $3 n^{2} + n$ . Find the first term and the common difference of the A.P. Hence, find its 15th term.

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

The sum of the first one term is the first term itself, so put n = 1.

Hint 2

Find $S_{2}$ ; then the second term is $S_{2} - S_{1}$ .

Hint 3

Use $a_{n} = a + (n - 1) d$ with n = 15.

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.

$S_{n} = 3 n^{2} + n$ . First term $a_{1} = S_{1} = 3 (1)^{2} + 1 = 4$
1 mark
$S_{2} = a_{1} + a_{2} = 3 (2)^{2} + 2 = 14$
1 mark
$a_{2} = S_{2} - S_{1} = 14 - 4 = 10$
1 mark
Common difference $d = a_{2} - a_{1} = 10 - 4 = 6$
1 mark
$a_{15} = a + 14 d = 4 + 14 \times 6 = 4 + 84 = 88$
1 mark

Answer: First term = 4, common difference = 6, 15th term = 88

Where the marks are

$a_{1} = S_{1} = 4$ (1 mark)
$S_{2} = 14$ (1 mark)
$a_{2} = 10$ (1 mark)
$d = 6$ (1 mark)
$a_{15} = 4 + 14 (6) = 88$ (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.

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

Hints

Step-by-step solution

Revise the chapter

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