Given:
The sum of the first 15 terms of an arithmetic sequence is 480.
To find:
The 8th term of the sequence.
Solution:
We have,
[tex]S_{15}=480[/tex]
We know that, the nth term of an AP is
[tex]a_n=a+(n-1)d[/tex]
Where, a is the first term and d is the common difference.
Putting n=8, we get the 8th term of the sequence.
[tex]a_8=a+(8-1)d[/tex]
[tex]a_8=a+7d[/tex]
The sum of first n terms is
[tex]S_{n}=\dfrac{n}{2}[2a+(n-1)d][/tex]
Where, a is the first term and d is the common difference.
[tex]S_{15}=\dfrac{15}{2}[2a+(15-1)d][/tex]
[tex]480=\dfrac{15}{2}[2a+14d][/tex]
[tex]480=\dfrac{15}{2}[2(a+7d)][/tex]
[tex]480=15(a+7d)[/tex]
Divide both sides by 15.
[tex]\dfrac{480}{15}=a+7d[/tex]
[tex]32=a+7d[/tex]
[tex]32=a_8[/tex]
Therefore, the 8th term of the arithmetic sequence is 32.
