Question

Solve: $\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7.....}}}}}=k$. Find k.

Answer 1

It is given that,

$\sqrt{7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots }}}}}=k$

Squaring both sides,

$7\sqrt{7\sqrt{7\sqrt{7\sqrt{7\cdots }}}}=k^2$

$7k=k^2$

$k^2-7k=0$

$k\left(k-7\right)=0$

$k=0ork=7$

The value $k=0$ cannot be possible, so, the only value is $k=7$.