**Q) 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 3 ^{rd} layer from the top there are 9 jars and so on till the 8^{th} 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} .**

**OR**

**(iii)**

**(B) The shopkeeper added 3 jars in each layer. How many jars are there in thhttps://youtu.be/J6EheU4L6Dwe 5**

^{th}layer from the top?**Ans:**

**VIDEO SOLUTION**

**STEP BY STEP SOLUTION**

**(i) AP and common difference:**

Since there are 3 jars in the top layer and 6 jars in 2^{nd} layer, 9 jars in 3^{rd} layers and so on …

Since the terms (number of jars) are increasing at regular gap in each layer, hence, it forms an AP.

Its first term, a = number of jars in 1^{st} layer = 3

and AP is formed as: 3, 6, 9, …..

Since the common difference is difference in values of any two consecutive rows

In above AP, a_{1 }= 3 and a_{2} = 6

∴ common difference d for the given AP = a_{2} – a_{1} = 6 – 3 = 3

**Therefore, AP is 3, 6, 9, ….. and the common difference is 3**

**(ii) Possibility of 34 jars in a layer:**

If 34 jars are kept in a layer, it should be a value of a term of our AP: 3, 6, 9,….

Let’s consider 34 is the value of n^{th} term in our AP.

Now, we know that the value of n^{th} term of an AP is given by:

T_{n} = a + (n – 1) d

by substituting, T_{n} = 34, a = 3 and d = 3 in the above expression, we get:

∴ 34 = 3 + (n – 1) 3

∴ 34 = 3 + 3 n – 3

∴ 34 = 3 n

∴ n =

Since to be a term of an AP, value of n has to be a natural number which is not possible in this case.

**Therefore, 34 jars in a layer are not possible in this pattern.**

**(iii) Expression for S _{n} and Value of S_{8}:**

In the given pattern, number of jars are forming value of a term and we are given n number of rows.

Since, in an AP, if there are n terms, then the total of the n terms of AP is given by:

Sn = [2 a + (n – 1) d]

∴ for a = 3 and d = 3, Sn = [2 (3) + (n – 1) (3)]

∴ Sn = [6 + (n – 1) 3]

∴ Sn = [3 n + 3]

∴ Sn = [n + 1]

**This is our expression for total number of items in n rows.**

Now, let’s calculate value of S_{8 }by putting n = 8:

∵ Sn = [n + 1]

∴ S_{8} = [8 + 1]

∴ S_{8} = 12 x 9 = 108

**Therefore the value of S _{8} is 108**

**(iii)(B) Number of jars in 5 ^{th} layer in new pattern:**

Since Shopkeeper added 3 jars in each layer, hence

Number of jars in 1^{st} (top) row = 3 + 3 = 6 i.e. revised a_{1} = 6

Number of jars in 2nd layer = 6 + 3 = 9 i.e. revised a_{2} = 9

New AP = 6, 9, 12,…..

and common difference = a_{2} – a_{1} = 9 – 6 = 3 i.e. d = 3

Let’s calculate number of jars in 5^{th} row and this will be given by value of 5^{th} term of our new AP.

Now, the value of n^{th} term of an AP is given by:

T_{n}= a + (n – 1) d

Hence, value of 5^{th} term with n = 5, a = 6 and d = 3:

∴ T_{5 }= 6 + (5 – 1) 3

= 6 + 12 = 18

**Therefore, there will be 18 jars in 5 ^{th} layer in the new pattern.**

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