Question

The third term of an A.P. is 7 and the seventh term exceeds three times the third term by 2. Find the first term, the common difference and the sum of first 20 terms.

Answer 1

Given,
$a_3=7=a_7-3a_3=2$
We have:
$a_3=7\Rightarrow a+(3-1)d=7\Rightarrow a+2d=7\dots \dots \left(i\right)$
Also, $a_7-21=2$ (Given)
$\Rightarrow a+(7-1)d=23\Rightarrow a+6d=23\dots \dots \left(ii\right)$
From (i) and (ii), we get:
4d = 16
$\Rightarrow d=4$
Putting the value in (i), we get:
$a+2\left(4\right)=7\Rightarrow a=-1\therefore S_{20}=\frac{20}{2}\left[2\right(-1)+(20-1\left)\right(4\left)\right]\Rightarrow S_{20}=10[-2+76]\Rightarrow S_{20}=10\left[74\right]=740\therefore a=-1,d=4,S_{20}=740$