Question

Let $S_n$ denotes the sum of first $n$ terms of an AP. If $S_4=-34,S_5=-60$ and $S_6=-93$, then the common difference and the first term of the AP are respectively.

• A. $-7,2$
• B. $7,-4$
• C. $7,-2$
• D. $-7,-2$
• E. $4,-7$

Answer 1

The correct option is A $-7,2$

Given, $S_n=$ Sum of first $n$ terms of an AP
Let $a$ and $d$ be the first term and common difference of an AP.
Therefore, $S_4=\frac{4}{2}[2a+(4-1\left)d\right]=-34$
$(\because S_n=\frac{n}{2}[2a+(n-1)d\left]\right)$
$\Rightarrow 2a+3d=-17....\left(i\right)$
and $S_5=\frac{5}{2}[2a+(5-1\left)d\right]=-60$
$\Rightarrow 2a+4d=-24....\left(ii\right)$
On subtracting Eq. (i) from Eq. (ii), we get
$d=-7$
Then, from Eq. (i), $2a-21=-17$
$\Rightarrow 2a=4\Rightarrow a=2$
Hence, common difference, $d=-7$ and first term $=2$.