Let's break down each of these problems:

1. Find the sum of $5, -5, 5, -5$:

This sequence alternates between $+5$ and $-5$. The sum is: $5 + (-5) + 5 + (-5) = 0$ The sum of these four terms is 0.

2. Find the sum of $\frac{1}{9}, \frac{1}{27}, \frac{1}{61}, \frac{1}{243}$:

This is the sum of four fractions: $\frac{1}{9} + \frac{1}{27} + \frac{1}{61} + \frac{1}{243}$ To add these, we need to find the least common denominator (LCD). The LCD of these fractions is quite large due to the different denominators, so let's compute:

$\frac{1}{9} + \frac{1}{27} + \frac{1}{61} + \frac{1}{243} = \frac{540}{4761} + \frac{180}{4761} + \frac{78}{4761} + \frac{20}{4761} = \frac{818}{4761}$ The sum is $\frac{818}{4761}$, which is approximately 0.1717.

3. Find the sum of the first 12 terms in $\frac{1}{9}, -\frac{1}{9}, \frac{1}{9}, -\frac{1}{9}, \dots$:

This is an alternating series where each term alternates between $\frac{1}{9}$ and $-\frac{1}{9}$. The sum of the first 12 terms is: $\left( \frac{1}{9} + (-\frac{1}{9}) \right) + \left( \frac{1}{9} + (-\frac{1}{9}) \right) + \dots$ Since each pair sums to 0: $0 + 0 + \dots = 0$ The sum of the first 12 terms is 0.

4. Find the sum of the first 15 terms in $5, -5, 5, -5, \dots$:

This is also an alternating series. Notice that every pair of terms sums to $5 + (-5) = 0$. Since there are 15 terms, there are 7 complete pairs and one extra term: $(5 + (-5)) + (5 + (-5)) + \dots + 5 = 0 + 0 + \dots + 5 = 5$ The sum of the first 15 terms is 5.

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Would you like more details on any of these calculations or have any questions?

Here are some related questions to consider:

1. What is the sum of the first 10 terms in the sequence $3, -3, 3, -3, \dots$?
2. What is the sum of $\frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \frac{1}{16}$?
3. How do you find the sum of an alternating infinite geometric series?
4. What is the sum of the first 20 terms in the sequence $7, -7, 7, -7, \dots$?
5. Can the sum of fractions always be simplified to a simpler form?

Tip: When dealing with alternating series, look for patterns to simplify the calculation, especially when terms cancel out.