Ms. Sheela visited a store near her house and found that the glass jars are arranged one above the other in a specific pattern. On the top layer there are 3 jars. In the next layer there are 6 jars. In the 3rd layer from the top there are 9 jars and so on till the 8th layer. On the basis of the above situation answer the following questions. (i) Write an A.P whose terms represent the number of jars in different layers starting from top. Also, find the common difference. (ii) Is it possible to arrange 34 jars in a layer if this pattern is continued? Justify your answer. (iii)(A) If there are 'n' number of rows in a layer then find the expression for finding the total number of jars in terms of n. Hence find S_8. (iii)(B) The shopkeeper added 3 jars in each layer. How many jars are there in the 5th layer from the top?

MathsCBSE Class 10 · Standard · Sample paper, session 2024-25 · 4 marks · Case-based

CBSE Class 10 Mathematics Standard sample paper 2024-25: question 36

From the official CBSE sample question paper for 2024-25, section E, question 36, for 4 marks. Try it first; the hints and our step-by-step solution are below.

Q36 · 4 marks · Case-basedCompetency-basedArithmetic Progressions

Ms. Sheela visited a store near her house and found that the glass jars are arranged one above the other in a specific pattern. On the top layer there are 3 jars. In the next layer there are 6 jars. In the 3rd layer from the top there are 9 jars and so on till the 8th layer.

On the basis of the above situation answer the following questions.

(i)
Write an A.P whose terms represent the number of jars in different layers starting from top. Also, find the common difference.
1 mark
(ii)
Is it possible to arrange 34 jars in a layer if this pattern is continued? Justify your answer.
1 mark
(iii)(A)
If there are ' $n$ ' number of rows in a layer then find the expression for finding the total number of jars in terms of $n$ . Hence find $S_{8}$ .
2 marks
OR
(iii)(B)
The shopkeeper added 3 jars in each layer. How many jars are there in the 5th layer from the top?
2 marks

Source: official sample paper from cbseacademic.nic.in (PDF). See it in the full Sample paper 2024-25.

Hints

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

Hint 1

List the jars layer by layer: 3, 6, 9, … What is added each time?

Hint 2

To check whether 34 can appear, solve $a + (n - 1) d = 34$ and see whether $n$ is a whole number.

Hint 3

For the sum use $S_{n} = \frac{n}{2} [2 a + (n - 1) d]$ . If 3 jars are added to every layer, only the first term changes.

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 jars in the layers from the top form the A.P. $3, 6, 9, 12, \dots, 24$ (8 layers).
½ mark
(i) Common difference $d = 6 - 3 = 3$ .
½ mark
(ii) Suppose a layer has 34 jars. Using $a_{n} = a + (n - 1) d$ :
$34 = 3 + (n - 1) \times 3$
$3 n = 34$ , so $n = \frac{34}{3} = 11 \frac{1}{3}$ .
½ mark
(ii) $n$ is not a positive integer. So 34 is not a term of the A.P., and a layer of 34 jars is not possible in this pattern.
½ mark
(iii)(a) Take $n$ as the number of layers. Here $a = 3$ and $d = 3$ .
$S_{n} = \frac{n}{2} [2 a + (n - 1) d] = \frac{n}{2} [2 \times 3 + (n - 1) 3]$
½ mark
(iii)(a) Simplifying:
$S_{n} = \frac{n}{2} [6 + 3 n - 3] = \frac{n}{2} (3 n + 3)$
$S_{n} = \frac{3 n ( n + 1 )}{2}$
1 mark
(iii)(a) $S_{8} = \frac{3 \times 8 \times 9}{2} = 108$ jars.
½ mark
(iii)(b) After adding 3 jars to each layer, the A.P. becomes $6, 9, 12, \dots$ with $a = 6$ and $d = 3$ .
½ mark
(iii)(b) $a_{5} = a + 4 d = 6 + 4 \times 3$
1 mark
(iii)(b) $a_{5} = 18$ jars.
½ mark

Answer: (i) 3, 6, 9, …, 24; d = 3; (ii) Not possible, since n = 34/3 is not a natural number; (iii)(a) Sₙ = 3n(n+1)/2, S₈ = 108; (iii)(b) 18 jars

Where the marks are

(i) Writes the A.P. 3, 6, 9, … (½ mark)
(i) Gets $d = 3$ (½ mark)
(ii) Forms $34 = 3 + (n - 1) 3$ and gets $n = 34/3$ (½ mark)
(ii) Concludes it is not possible because $n$ is not a natural number (½ mark)
(iii)(a) Writes the sum formula with $a = 3$ , $d = 3$ (½ mark)
(iii)(a) Simplifies to $S_{n} = 3 n (n + 1) /2$ (1 mark)
(iii)(a) Gets $S_{8} = 108$ (½ mark)
(iii)(b) Identifies the new A.P. with $a = 6$ , $d = 3$ (½ mark)
(iii)(b) Uses $a_{5} = 6 + 4 \times 3$ (1 mark)
(iii)(b) Gets 18 jars (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 paperSample paper 2024-25 (Standard): every question, by section.

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CBSE Class 10 Mathematics Standard sample paper 2024-25: question 36

Hints

Step-by-step solution

Revise the chapter

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