The Sum to 500 of a G.P is 15-1/10 and the first term is -1/8 find the common ratio. The fifth term of a G.P is 2/81 if the first term is 2 find the common ratio and Sum of the first 5 terms of the G.P.

The formula is $a_n = ar^{n-1}$, where $a$ is the first term and $r$ is the common ratio.

Given $a_5 = \frac{2}{81}$, $a = 2$, and $n=5$.

Divide both sides by 2:

Take the fourth root of both sides:

Since the problem does not specify if the terms are positive or negative, both are mathematically possible. However, typically for such problems, the positive root is assumed unless otherwise stated. Let's assume $r = \frac{1}{3}$ for the sum calculation. If the problem intended a specific sign, it would usually be mentioned.

The formula is $S_n = \frac{a(1-r^n)}{1-r}$ (for $r \neq 1$).

Given $a = 2$, $n = 5$. Let's use $r = \frac{1}{3}$.

First, calculate $\left(\frac{1}{3}\right)^5$:

Now substitute this back into the sum formula:

To divide by a fraction, multiply by its reciprocal:

If we used $r = -\frac{1}{3}$:

Given the context of typical problems, the positive common ratio is usually intended.