Question

sum n=3 ^ infty 6 ( 2 3 )^ n-3

Answer 1

We need to evaluate this:

Let me attach the work:

You can see that the series is infinite geometric series.

The first term (a) would be:

$\frac{2}{3}^3=\frac{8}{27}$

The common ratio r is 2/3

So, sum to inifnity would be:

$\text{Sum}=\frac{a}{1-r}=\frac{\frac{8}{27}}{1-\frac{2}{3}}=\frac{8}{9}$

Therefore:

$\frac{81\times(\frac{8}{9})}{4}$$=18$

Answer is 18