‘Kolam’ is a decorative art which is made with rice flour in South Indian States. It is drawn on grid pattern of dots. One such art work is shown below. Observe the given figure carefully. There are 4 dots in first square, 8 dots in second square, 12 dots in third square and so on. Based on the above, answer the following questions : (i) Show that number of dots given above form an A.P. Write the first term and common difference. (ii) Write n^th term of the A.P. formed. (iii)(a) The pattern is expanded on a large ground. If total 220 dots are used, then find the number of squares formed. (iii)(b) Is it possible to complete n number of squares using 100 dots ? If yes, then find the value of n.

MathsCBSE Class 10 · Standard · Board exam 2026, set 30/4/1 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard 2026: question 37

Set in the CBSE Class 10 board exam of 2026, paper 30/4/1, section E, question 37, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q37 · 4 marks · Case-basedCompetency-basedArithmetic Progressions

‘Kolam’ is a decorative art which is made with rice flour in South Indian States. It is drawn on grid pattern of dots. One such art work is shown below.

Observe the given figure carefully. There are 4 dots in first square, 8 dots in second square, 12 dots in third square and so on.

Based on the above, answer the following questions :

(i)
Show that number of dots given above form an A.P. Write the first term and common difference.
1 mark
(ii)
Write $n^{t h}$ term of the A.P. formed.
1 mark
(iii)(a)
The pattern is expanded on a large ground. If total 220 dots are used, then find the number of squares formed.
2 marks
OR
(iii)(b)
Is it possible to complete n number of squares using 100 dots ? If yes, then find the value of n.
2 marks

Figure in the paper: A Kolam pattern drawn on a diamond-shaped grid of dots: concentric square rings of dots (rotated 45°) around a central black filled square, with looping curves around the dots. See it in the official paper.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2026 board paper, set 30/4/1.

Hints

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

Hint 1

Check the difference between consecutive terms 4, 8, 12.

Hint 2

Total dots means the sum of n terms: Sₙ = n/2 [2a + (n − 1)d].

Hint 3

Solve the quadratic in n; it is possible only if n turns out to be a natural number.

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.

(i) The numbers of dots are 4, 8, 12, …
Here $8 - 4 = 12 - 8 = 4$ . The difference is constant, so the numbers form an A.P.
½ mark
First term $a = 4$ and common difference $d = 4$ .
½ mark
(ii) $a_{n} = a + (n - 1) d = 4 + (n - 1) 4 = 4 n$
1 mark
(iii)(a) Use $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ with $S_{n} = 220$ :
$220 = \frac{n}{2} [8 + 4 (n - 1)]$
½ mark
$\Rightarrow 220 = 2 n (n + 1) \Rightarrow n^{2} + n - 110 = 0$
½ mark
$\Rightarrow (n + 11) (n - 10) = 0 \Rightarrow n = 10$ or $n = - 11$
$n$ cannot be negative, so $n = 10$ . Hence 10 squares are formed.
1 mark
(iii)(b) Take $S_{n} = 100$ :
$100 = \frac{n}{2} [8 + 4 (n - 1)] = 2 n (n + 1)$
½ mark
$\Rightarrow n^{2} + n - 50 = 0$
½ mark
$n = \frac{- 1 \pm 1 + 200}{2} = \frac{- 1 \pm 201}{2}$
$201$ is not an integer, so $n$ is not a natural number. Therefore it is not possible to complete squares using exactly 100 dots.
1 mark

Answer: (i) AP with a = 4, d = 4; (ii) aₙ = 4n; (iii)(a) 10 squares; (iii)(b) Not possible, since n = (−1 ± √201)/2 is not a natural number

Where the marks are

(i) Shows the common difference 4 is constant, so the dots form an A.P. (½ mark)
(i) States a = 4, d = 4 (½ mark)
(ii) aₙ = 4n (1 mark)
(iii)(a) Correct substitution in the Sₙ formula with Sₙ = 220 (½ mark)
(iii)(a) Forms n² + n − 110 = 0 (½ mark)
(iii)(a) Solves to get n = 10 squares, rejecting −11 (1 mark)
(iii)(b) Correct substitution with Sₙ = 100 (½ mark)
(iii)(b) Forms n² + n − 50 = 0 (½ mark)
(iii)(b) Shows n is not natural, so it is not possible (1 mark)

Revise the chapter

Chapter: Arithmetic Progressions (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Arithmetic Progressions, one at a time, marked against the answer key and marking points.Back to the full paper2026 board paper, set 30/4/1 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard 2026: question 37

Hints

Step-by-step solution

Revise the chapter

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