Split the summation to make the starting value of equal to .

The formula for the geometric series is:

Substitute the values into the formula and make sure to multiply by the front term.

Simplify.

Simplify the numerator.

Raise to the power of .

Multiply by .

Subtract from .

Reduce the expression by cancelling the common factors.

Subtract from .

Cancel the common factor of .

Factor out of .

Cancel the common factor.

Rewrite the expression.

Multiply by .

Expand the series for each value of .

Simplify.

Simplify each term.

Subtract from .

Anything raised to is .

Multiply by .

Subtract from .

Evaluate the exponent.

Multiply by .

Add and .

Replace the summations with the values found.

Subtract from .

Evaluate sum from n=3 to 12 of 20(0.5)^(n-1)