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Q.
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
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 Eq. (i) from Eq. (ii), we get
$d=-7$
Then, from Eq. (i), $2 a-21=-17$
$\Rightarrow \, 2 a=4$
$ \Rightarrow a=2$
Hence, common difference, $d=-7$ and first term $=2$