Question

Find the common difference of an AP. whose first term is $100$ and the sum of whose first six terms is five times the sum of the next six terms.

• A. $10$
• B. $-10$
• C. $5$
• D. $-5$

Answer 1

The correct option is B $-10$

Here $a=100$

Let difference is $d$.
$\Rightarrow a_1+a_2+a_3+a_4+a_5+a_6=5(a_7+a_8+a_9+a_{10}+a_{11}+a_{12})$

So by the formula, $S_n=\frac{n}{2}(a+l)$, where $a$ &$l$ are the first and last term of an AP, we have
$6\left(\frac{a_1+a_6}{2}\right)=5\times 6\left(\frac{a_7+a_{12}}{2}\right)$

$\Rightarrow a_1+a_6=5(a_7+a_{12})$

$\Rightarrow a+a+5d=5(a+6d+a+11d)$

$\Rightarrow 2a+5d=10a+85d$

$\Rightarrow 80d=-8a$

$\Rightarrow d=\frac{-a}{10}\Rightarrow \frac{-100}{10}\Rightarrow -10$