Question

Find the sum of first $20$ terms of an A.P., in which $3^{rd}$ term is $7$ and $7^{th}$ term is two more than thrice of its $3^{rd}$ term.

Answer 1

Let $a$ be the first term and $d$ be the common difference of the given A.P.

Then, $a_3=7$ and $a_7=3a_3+2$

$⟹a+2d=7$ and $a+6d=3(a+2d)+2$
$⟹a+2d=7$ and $a+6d=3(a+2d)+2$
$⟹a+2d=7$ and $a=-1$
$⟹a=-1$ and $d=4$
Putting $n=20,a=-1$ and $d=4$ in $S_n=\frac{n}{2}\{2a+(n-1\left)d\right\}$, we get
$S_{20}=\frac{20}{2}\{2\times -1+(20-1)\times 4\}=\frac{20}{2}(-2+76)=740$