Question

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:

1. Write an A.P. whose terms represent the number of jars in different layers starting from top. Also, find the common difference.
2. Is it possible to arrange 34 jars in a layer if this pattern is continued? Justify your answer.
3. 1. 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 S₈.
                                       OR
   2. The shopkeeper added 3 jars in each layer. How many jars are there in the 5^th layer from the top?

Answer 1

i. Thus, the jars form an A.P. 3, 6, 9, ..... up to 8 terms.

This is an A.P. with

First term (a) = 3

Common difference (d) = 6 − 3 = 3

ii. Now, A.P. is 3, 6, 9, ..... upto 8 terms.

Let,

a_n = 34

a + (n − 1)d = 34

Putting a = 3, d = 3

3 + (n − 1) × 3 = 34

3 + 3n − 3 = 34

3n = 34

n = `34/3`

n = 11.33

Since n is the number of terms, it cannot be in decimals.

∴ 34 is not a term in A.P.

It is not possible to arrange 34 jars in a layer.

iii. A. Now, A.P. is 3, 6, 9, ... up upto 8 terms.

Now,

S_n = `n/2 [2a + (n − 1) d]`

Putting a = 3, d = 3

= `n/2 [2(3) + (n − 1) 3]`

= `n/2 [6 + 3n − 3]`

= `n/2 [3 + 3n]`

Thus, S_n = `n/2 [3 + 3n]`

Putting n = 8 in S_n

S₈ = `8/2 [3 + 3 × 8]`

= 4[3 + 24]

= 4 × 27

= 108

OR

B. Previous A.P. was 3, 6, 9, ....

After adding 3 jars in each layer, A.P. is

3 + 3, 6 + 3, 9 + 3, ….

6, 9, 12, …

So,

First term (a) = 6

Common difference (d) = 9 − 6 = 3

Now,

a_n = a + (n − 1) d

Putting n = 5, a = 6, d = 3

= 6 + (5 − 1) × 3

= 6 + 4 × 3

= 6 + 12

= 18

Thus, there are 18 jars in the 5^th layer.