Question

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

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

Answer 1

Answer: (A)

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.
$\therefore S_4=\frac{4}{2}[2a+(4-1)d]=-34$
$\left(\because S_n=\frac{n}{2}[2a+(n-1)d]\right)$
$\Rightarrow 2a+3d=-17\dots (i)$
and $S_5=\frac{5}{2}[2a+(5-1)d]=-60$
$\Rightarrow 2a+4d=-24\dots (ii)$
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 $=2$