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

$S_n=\dfrac{n}{2}[2a+(n-1)d]$

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

$292.5=\dfrac{15}{2}[2(2)+(15-1)d]$

$292.5=\dfrac{15}{2}[4+14d]$

$292.5=15[2+7d]$

Divide both sides by 15.

$\dfrac{292.5}{15}=2+7d$

$19.5=2+7d$

$19.5-2=7d$

$17.5=7d$

Dividing both sides by 7, we get

$\dfrac{17.5}{7}=d$

$2.5=d$

Therefore, the common difference is 2.5.