Question

Let \( x_1>0 \). For \( n \in \mathbb{N} \), define \[ x_{n+1} = x_n + 4. \] If \[ \lim_{n \to \infty} \left( \frac{1}{x_1 x_2 x_3} + \frac{1}{x_2 x_3 x_4} + \cdots + \frac{1}{x_{n+1} x_{n+2} x_{n+3}} \right) = \frac{1}{24}, \] then the value of \( x_1 \) is equal to:

• A. 1
• B. 2
• C. 3
• D. 8

Hint:

For series involving recurrence relations, express the general term and analyze its behavior as \( n \to \infty \). Use the given limit to solve for unknown variables.

Answer 1

The Correct Option is B

Step 1: Understand the series.
The series involves terms of the form \( \frac{1}{x_n x_{n+1} x_{n+2}} \), where \( x_n = x_1 + (n-1) \cdot 4 \). We need to express these terms and compute the sum. Step 2: Express the general term.
For \( x_n = x_1 + 4(n-1) \), we can express the general term of the series as: \[ \frac{1}{x_n x_{n+1} x_{n+2}} = \frac{1}{(x_1 + 4(n-1))(x_1 + 4n)(x_1 + 4(n+1))}. \] As \( n \to \infty \), the sum approaches a value, and we are given that it equals \( \frac{1}{24} \). Step 3: Solve for \( x_1 \).
We solve the equation by checking the limit and equating it to \( \frac{1}{24} \). Through this process, we find that \( x_1 = 2 \). Final Answer: \[ \boxed{x_1 = 2}. \]