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: [tex]a=2[/tex]
Sum of first 15 terms: [tex]S_{15}=292.5[/tex]
The formula of sum of first n terms of an AP is
[tex]S_n=\dfrac{n}{2}[2a+(n-1)d][/tex]
Where, a is first term and d is common difference.
Putting [tex]S_{15}=292.5[/tex], n=15 and a=2 in the above formula, we get
[tex]292.5=\dfrac{15}{2}[2(2)+(15-1)d][/tex]
[tex]292.5=\dfrac{15}{2}[4+14d][/tex]
[tex]292.5=15[2+7d][/tex]
Divide both sides by 15.
[tex]\dfrac{292.5}{15}=2+7d[/tex]
[tex]19.5=2+7d[/tex]
[tex]19.5-2=7d[/tex]
[tex]17.5=7d[/tex]
Dividing both sides by 7, we get
[tex]\dfrac{17.5}{7}=d[/tex]
[tex]2.5=d[/tex]
Therefore, the common difference is 2.5.
