Question

Find the sum of:

$8+4\sqrt{2}+4+2\sqrt{2}+.....+\infty$

Answer 1

The equation you can use for the sum of the infinite sequence is:

$\sum a_i=\frac{a}{1-r}$

$a=8,r=\frac{1}{\sqrt{2}}$

$\therefore \sum a_i=\frac{8}{1-\frac{1}{\sqrt{2}}}$

$\sum a_i=\frac{8\sqrt{2}}{\sqrt{2}-1}$

$=\frac{8\sqrt{2}}{\sqrt{2}-1}\times \frac{\sqrt{2}+1}{\sqrt{2}+1}$

$=\frac{8\sqrt{2}(\sqrt{2}+1)}{2-1}$

$=\frac{16+8\sqrt{2}}{1}$

$\therefore \sum a_i=16+8\sqrt{2}$