Given:
The sum of third and seventh terms of an AP is 20.
To find:
The sum of the first nine
terms.
Solution:
We have, the sum of third and seventh terms of an AP is 20.
[tex]a_3+a_7=20[/tex] ...(i)
nth term of an AP is
[tex]a_n=a+(n-1)d[/tex]
where, a is first term and d is common difference.
[tex]a_3=a+(3-1)d[/tex]
[tex]a_3=a+2d[/tex] ...(ii)
[tex]a_7=a+(7-1)d[/tex]
[tex]a_7=a+6d[/tex] ...(iii)
Using (i), (ii) and (iii), we get
[tex](a+2d)+(a+6d)=20[/tex]
[tex]2a+8d=20[/tex] ...(iv)
Now, the sum of first n terms of an AP is
[tex]S_n=\dfrac{n}{2}[2a+(n-1)d][/tex]
Put n=9 to find the sum of first 9 terms of an AP.
[tex]S_n=\dfrac{9}{2}[2a+(9-1)d][/tex]
[tex]S_n=\dfrac{9}{2}[2a+8d][/tex]
[tex]S_n=\dfrac{9}{2}[20][/tex] [Using (iv)]
[tex]S_n=\dfrac{180}{2}[/tex]
[tex]S_n=90[/tex]
Therefore, the sum of the first 9 terms is 90.
