Consider the geometric series with first term -4 and common ratio -1/2. (a) Write the general term. (b) Write the series in sigma notation. (c) Calculate the sum to infinity.

The first term is $a = -4$.
The common ratio is $r = \frac{2}{-4} = -\frac{1}{2}$.

The general term is $\boxed{\mathbf{T_n = -4\left(-\frac{1}{2}\right)^{n-1}}}$.

3.1.2 Write the series in sigma notation.

The last term is $\frac{1}{32}$. Set $T_n = \frac{1}{32}$:

Since $128 = 2^7$, we can write:

There are 8 terms in the series.

The series in sigma notation is $\boxed{\mathbf{\sum_{k=1}^{8} -4\left(-\frac{1}{2}\right)^{k-1}}}$.

3.1.3 Calculate the sum to infinity of this series.

The common ratio is $r = -\frac{1}{2}$. Since $|r| = \left|-\frac{1}{2}\right| = \frac{1}{2} < 1$, the sum to infinity exists.

The sum to infinity is $\boxed{\mathbf{-\frac{8}{3}}}$.

3.2.1 Determine $S_5$.
Given the formula for the sum of $n$ terms: $S_n = 32 - 32\left(\frac{1}{2}\right)^n$.
Substitute $n=5$ into the formula:

The value of $S_5$ is $\boxed{\mathbf{31}}$.

3.2.2 How many terms must be added for the sum to be equal to $\frac{255}{8}$?
Set $S_n = \frac{255}{8}$:

Since $256 = 2^8$, we have:

The number of terms that must be added is $\boxed{\mathbf{8}}$.

3.2.3 If $2^n = p$, determine the value of $S_{5-n} - S_{5+n}$ in terms of $p$.

Given $2^n = p$, so $S_n = 32 - \frac{32}{p}$.

Substitute $2^n = p$:

Substitute $2^n = p$:

The value of $S_{5-n} - S_{5+n}$ in terms of $p$ is $\boxed{\mathbf{\frac{1 - p^2}{p}}}$.