Question

the first term of an arithmetic sequence is 2 and the sum of the first 15 terms is 292.5. Find the common difference

Answer 1

Given:

First term of an arithmetic sequence is 2.

Sum of first 15 terms = 292.5

To find:

The common difference.

Solution:

We have,

First term:
`a=2`

Sum of first 15 terms:
`S_(15)=292.5`

The formula of sum of first n terms of an AP is

`S_n=(n)/(2)[2a+(n-1)d]`

Where, a is first term and d is common difference.

Putting
`S_(15)=292.5`, n=15 and a=2 in the above formula, we get

`292.5=(15)/(2)[2(2)+(15-1)d]`

`292.5=(15)/(2)[4+14d]`

`292.5=15[2+7d]`

Divide both sides by 15.

`(292.5)/(15)=2+7d`

`19.5=2+7d`

`19.5-2=7d`

`17.5=7d`

Dividing both sides by 7, we get

`(17.5)/(7)=d`

`2.5=d`

Therefore, the common difference is 2.5.