https://file.aiquickdraw.com/mathbot/file/1726678031577ovpoo8b8.jpg

The image you uploaded contains two summation series problems, labeled as parts (a) and (c). Here’s a breakdown of what I can interpret from it:

### Problem (a):
The series looks like:

\[
1 \cdot 2 + 2 \cdot 3 + 3 \cdot 4 + \dots + n \cdot (n + 1)
\]

The formula provided is:

\[
\frac{n(n+1)(n+2)}{3}
\]

This seems to be the closed form for the sum of the series. It simplifies the summation of the products of consecutive integers.

### Problem (c):
This series appears as:

\[
1 \cdot 4 + 2 \cdot 7 + 3 \cdot 10 + \dots + n \cdot (3n + 1)
\]

And the closed form given is:

\[
n(n + 1)^2
\]

This is the compact form for the sum of the given series, which involves terms increasing with a factor of 3.

Would you like a detailed explanation for either of these problems? Let me know if you have any questions!

Here are five related questions:
1. How would you derive the closed form for problem (a) step by step?
2. Can you find a pattern for a similar series, such as \(1 \cdot 3 + 2 \cdot 5 + 3 \cdot 7 + \dots\)?
3. How can you generalize the sum of a series involving products of two terms?
4. What is the meaning of the factor \( \frac{1}{3} \) in problem (a)'s closed form?
5. Can you apply similar techniques to solve summations involving higher powers?

**Tip:** Always check for patterns in summation series by expressing a few terms and seeing if you can detect a recurrence relation or factor commonality!

Solve the summation series problems: 1. 1·2 + 2·3 + 3·4 + ... + n·(n+1) and 2. 1·4 + 2·7 + 3·10 + ... + n·(3n+1)

Learn how to solve summation series involving consecutive integers, including 1·2 + 2·3 + ... + n·(n+1) and 1·4 + 2·7 + ... + n·(3n+1). This guide provides detailed explanations and closed form solutions, suitable for high school students in Grades 10-12.

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