Question

Let $a_1,a_2,a_3$, ... be an arithmetic progression with nonzero common difference. It is given that ${\sum }_{i=4}^{12}{a}_i=63$ and $a_k=7$ for some k . Then the value of k is

• A. 6
• B. 7
• C. 8
• D. 9

Answer 1

Answer: (C)

Given, $a_1,a_2,a_3,...$ be an arithmetic progression
$a_1+a_2+a_3+...a_n=\frac{n}{2}\left(a_1+a_n\right)$
${\Sigma }_{i=4}^{12}{a}_i=63$
$\Rightarrow a_4+a_5+a_6+...+a_{12}=63$
$\Rightarrow \frac{9}{2}\left(a_4+a_{12}\right)=63$
$\Rightarrow a_4+a_{12}=14$
$\Rightarrow a+3d+a+11d=14$
$\Rightarrow 2a+14d=14$
$\Rightarrow a+7d=7$
$\Rightarrow a_8=7$
$\Rightarrow k=8$