Question

The sum of the $2nd$ and $7nd$ terms of an A.P. is $30$. If its $15th$ term is $1$ less than twice its $8th$ term, find the A.P.

Answer 1

A.P

Sum of $2^{nd}$ and $7^{th}$ term is 30

$a_{15}=2\times a_8-1$...(2)

$[\because a_n=a+(n-1\left)d\right]$ $a_2+a_7=30$

$a+(2-1)d+a(7-1)d=30$

$a+d+a+6d=30$

$2a+7d=30...\left(1\right)$

Now $a_{15}=2a_8-1$

$a+(15-1)d=2(a+(8-1\left)d\right)-1$

$a+14d=2a+7\times 2d-1$

$a+14d=2a+14d-1$

$a-2a=-1$

$-a=-1$

$a=1$

Now putting $a=1$ in eq${}^n$ (1), we get

$2\left(1\right)+7d=30$

$7d=28$

$d=4$

Thus the A.P is

$1,5,9,13,.......$

where $a=1$ & $d=4$