Question

The first term of an arithmetic progression is $1$ and the sum of the first nine terms is $369 .$ The first and the ninth term of a geometric progression coincide with the first and the ninth term of this arithmetic progression. The value of the seventh term of the geometric progression is______.

Answer 1

$369=\frac{9}{2}[2+(9-1) d]$
$\Rightarrow 82=2+8 d$
$\Rightarrow d=10$
Now, $a r^{8}=a+8 d$
$\Rightarrow 1 \times r^{8}=1+8 \times 10$
$\Rightarrow r^{8}=81$
$\Rightarrow r=\sqrt{3}$
$\Rightarrow a r^{(7-1)}=1 \times(\sqrt{3})^{6}=27$