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!