Question

Let $a_1,a_2,\dots ,a_{30}$ be in A.P., $S=\sum _{i=1}^{30}{a}_i$ and $T=\sum _{i=1}^{15}{a}_{2i-1}$. If $a_5=27$ and $S-2T=75$, then the value of $a_{10}$ is

• A. $47$
• B. $52$
• C. $42$
• D. $57$

Answer 1

The correct option is B $52$

Let $d$ be the common difference of A.P.
$S=a_1+a_2+\cdots +a_{30}=\frac{30}{2}[2a_1+29d]=15[2a_1+29d]$
$T=a_1+a_3+\cdots +a_{29}=\frac{15}{2}[2a_1+14(2d\left)\right]=15[a_1+14d]$

Given that $S-2T=75$
$\Rightarrow 15d=75\Rightarrow d=5$
Also given that $a_5=27$
$\Rightarrow a_1+4d=27\Rightarrow a_1=7$
Hence, $a_{10}=a_1+9d=52$