https://file.aiquickdraw.com/mathbot/file/1727307149293a7421ruo.jpg

Let's break down the four questions from your image and express them in summation notation as required:

### 12) \( 7 + 14 + 28 + \cdots + 896 \)
This is a geometric sequence where:
- First term \( a = 7 \)
- Common ratio \( r = 2 \)

The general term \( a_n \) of a geometric sequence is given by:
\[
a_n = a \cdot r^{n-1} = 7 \cdot 2^{n-1}
\]
The last term \( a_n = 896 \), so we can write this in summation notation:
\[
\sum_{n=1}^{N} 7 \cdot 2^{n-1}, \quad \text{where } N \text{ is the term number for } 896.
\]

To find \( N \), solve \( 7 \cdot 2^{N-1} = 896 \):
\[
2^{N-1} = \frac{896}{7} = 128 \quad \Rightarrow \quad 2^{N-1} = 2^7 \quad \Rightarrow \quad N-1 = 7 \quad \Rightarrow \quad N = 8
\]
Thus, the summation is:
\[
\sum_{n=1}^{8} 7 \cdot 2^{n-1}
\]

---

### 13) \( \frac{4}{5} + \frac{5}{6} + \frac{6}{7} + \cdots \)
This looks like a sequence of the form:
\[
\frac{n+3}{n+4}, \quad n = 1, 2, 3, \dots
\]
The general term can be written as \( \frac{n+3}{n+4} \), starting from \( n = 1 \).

Thus, the summation notation is:
\[
\sum_{n=1}^{\infty} \frac{n+3}{n+4}
\]
This is an infinite series, as no final term is given.

---

### 14) \( -1 + 2 - 3 + 4 - 5 + 6 - \cdots + 20 \)
This is an alternating series where the signs alternate between positive and negative for consecutive terms.

The general term can be expressed as \( (-1)^{n+1} \cdot n \), where \( n = 1, 2, 3, \dots \), and the last term is \( 20 \).

Thus, the summation notation is:
\[
\sum_{n=1}^{20} (-1)^{n+1} \cdot n
\]

---

### 15) \( 1 + \frac{1}{2} + \frac{2}{3} + \frac{6}{4} + \frac{24}{5} + \frac{120}{6} \)
This series follows a pattern where the numerators are factorials, i.e., \( 1!, 2!, 3!, 4!, 5!, 6! \).

Thus, the general term can be written as:
\[
\frac{(n-1)!}{n}
\]
where \( n = 1, 2, 3, \dots, 6 \).

So the summation notation is:
\[
\sum_{n=1}^{6} \frac{(n-1)!}{n}
\]

---

Would you like more details or clarifications on any of these summations?

### 5 Related Questions:
1. How do you find the sum of a geometric series?
2. What is the formula for the sum of an infinite series?
3. How do alternating series converge?
4. Can factorial terms be generalized for other sequences?
5. How would you find the sum of a finite arithmetic sequence?

### Tip:
When working with series, always check if it's geometric, arithmetic, or follows a unique pattern (like factorials) to simplify finding a closed form.

Use summation notation to write the sum: 7 + 14 + 28 + ... + 896, 4/5 + 5/6 + 6/7 + ..., -1 + 2 - 3 + 4 - 5 + 6 ... + 20, and 1 + 1/2 + 2/3 + 6/4 + 24/5 + 120/6.

Learn how to write sums in summation notation for various sequences, including a geometric sequence 7 + 14 + 28 + ... + 896, a series of fractions like 4/5 + 5/6 + ..., an alternating series -1 + 2 - 3 ..., and a factorial series like 1 + 1/2 + 2/3 + 6/4 ... + 120/6. This topic covers key algebraic and sequence concepts for high school students.

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