If the first term of a G.P is 10 and the sum of the infinite terms is 20. Find the sum of the first 6 terms of the G.P.

The first term is $a = 10$.
The sum of the infinite terms is $S_\infty = 20$.

The formula for the sum to infinity of a GP is $S_\infty = \frac{a}{1-r}$, provided that $|r| < 1$.
Substitute the given values into the formula:

To solve for $1-r$, multiply both sides by $(1-r)$ and divide by 20:

Now, solve for $r$:

Since $|r| = \left|\frac{1}{2}\right| < 1$, this value of $r$ is valid for an infinite sum.

The formula for the sum of the first $n$ terms of a GP is $S_n = \frac{a(1 - r^n)}{1 - r}$. We use this form because $|r| < 1$.
We have $a=10$, $r=\frac{1}{2}$, and we want to find $S_6$, so $n=6$.
Substitute these values into the formula:

First, calculate $\left(\frac{1}{2}\right)^6$:

Next, calculate the denominator $1 - \frac{1}{2}$:

Now substitute these back into the $S_6$ formula:

Calculate the term in the parenthesis:

Substitute this back:

To simplify, we can multiply by the reciprocal of the denominator:

Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 4:

The sum of the first 6 terms is $\boxed{\frac{315}{16}}$.