Mathematics - Sequence & Series Question with Solution | TestHub

Let common difference$=d$

$\therefore S= \sum _{i=1}^{30}{a}_i=\frac{30}{2}\left(2a_1+29 d\right)$

$\therefore S=15 \left(2a_1+29 d\right)$$........ \left(i\right)$

$T=\frac{15}{2}\left(2a_1+14 . 2d\right)$

$T=15 \left(a_1+14d\right)$$...........\left(ii\right)$

$a_1+4d=27$$......... \left(iii\right)$

$S-2T=75$

$\Rightarrow 15\left(2a_1+29 d\right)-30 \left(a_1+14d \right)=75$
$\Rightarrow 15 d\left(29-28\right)= 75$
$\Rightarrow d=5$
$\therefore$from$\left(iii\right)$$a_1=7$
$\therefore a_{10}=a_1+9d=7+45=52$