Find the sum of 5, -5, 5, -5

Find the sum of 1/9, 1/27, 1/61, 1/243

Find the sum of the first 12 terms in 1/9, -1/9, 1/9, -1/9, .....

Find the sum of the first 15 terms in 5, -5, 5, -5

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.

Find the sum of 5, -5, 5, -5; Find the sum of 1/9, 1/27, 1/61, 1/243; Find the sum of the first 12 terms in 1/9, -1/9, 1/9, -1/9, .....; Find the sum of the first 15 terms in 5, -5, 5, -5

Explore how to solve problems involving the sum of alternating series like 5, -5, 5, -5 and fractional series such as 1/9, 1/27, 1/61, 1/243. Detailed step-by-step explanations are provided, covering core arithmetic and fraction concepts. Ideal for students in Grades 6-8.

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