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

$a_3=7$ $a_7=3a_3+2$

$a_7=3\times 7+2=23$

$a+2d=7\to \left(1\right)$

$a+6d=23\to \left(2\right)$

$\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_$

$-4d=16$

$d=4\to \left(3\right)$

Sub $\left(3\right)$ in $\left(1\right)$

$a+2\times 4=7$

$a+8=7\Rightarrow a=-1$

$S_n=\frac{n}{2}(2a+(n-1\left)d\right)$

$S_{20}=\frac{20}{2}(2\times -1+(20-1\left)4\right)$

$S_{20}=10(76-2)$

$S_{20}=10\times 740$

$S_{20}=740$