7
2 × 3 × 4
+ 9
3 × 4 × 5
+ 11
4 × 5 × 6
+ ⋅ ⋅ ⋅ + 2n + 3
n(n + 1)(n + 2).

The expression you've provided appears to follow a specific pattern. Let's break it down step by step:

The general term in the sum is:

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

You have the sum of the series starting from \(n = 1\):

\[
S = \frac{7}{2 \times 3 \times 4} + \frac{9}{3 \times 4 \times 5} + \frac{11}{4 \times 5 \times 6} + \cdots + \frac{2n + 3}{n(n+1)(n+2)}
\]

To simplify and explore the sum further, observe that each term follows this general structure, which simplifies as follows:

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

This can be decomposed using partial fractions:

\[
\frac{2n + 3}{n(n+1)(n+2)} = \frac{A}{n} + \frac{B}{n+1} + \frac{C}{n+2}
\]

To determine the values of \(A\), \(B\), and \(C\), solve the system obtained by rewriting the left-hand side in terms of these fractions.

Would you like to go into the details of solving the partial fractions and evaluating the sum for a specific \(n\)?

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### 5 Related Questions:
1. How do you solve a partial fraction decomposition?
2. Can you find the closed form for this sum as \(n\) goes to infinity?
3. How do you analyze the convergence of an infinite series?
4. How does the general term of a series affect the behavior of the sum?
5. What are other examples of series that involve rational expressions?

### Tip:
Breaking down complex rational expressions using partial fractions can simplify calculations and help identify patterns in sums.

2 × 3 × 4 + 9 ÷ 3 × 4 × 5 + 11 ÷ 4 × 5 × 6 + ⋅ ⋅ ⋅ + 2n + 3 ÷ n(n + 1)(n + 2)

Explore the series with the general term (2n + 3) ÷ (n(n+1)(n+2)). Learn how to simplify the expression using partial fraction decomposition, understand the structure of the series, and analyze its convergence. Suitable for high school students in Grades 10-12.

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