Question

Let $S_n$ denote the sum of first n terms of an $A.P.$ If $S_4 = -3 4 , S_5 = -60$ 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

Answer 1

Answer: (A)

-7 ,2

Answer 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.
$\therefore\,S_{4}=\frac{4}{2}[2 a+(4-1) d]=-34$
$\left(\because S_{n}=\frac{n}{2}[2 a+(n-1) d]\right)$
$\Rightarrow\, 2 a+3 d=-17\,\,\,\,\,\dots(i)$
and $S_{5}=\frac{5}{2}[2 a+(5-1) d]=-60$
$\Rightarrow\, 2 a+4 d=-24\,\,\,\,\,\dots(ii)$
On subtracting E (i) from E (ii), we get
$d=-7$
Then, from E (i), $2 a-21=-17$
$\Rightarrow\, 2 a=4$
$\Rightarrow a=2$
Hence, common difference, $d=-7$ and first term $=2$