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

Answer: 27

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