Q.5. The terms $x_n$ of a sequence follow the rule

$$x_{n+1} = \frac{x_n + p}{x_n + q}$$

where $p$ and $q$ are real numbers.

Given that $x_1 = 3$, $x_2 = 5$, and $x_3 = 7$, find the value of $x_4$.

Answer Choices

• $-5$
• $5$
• $\frac{51}{7}$
• $\frac{15}{2}$
• $\frac{23}{3}$
• $9$
• $11$
• $13$

Solution

We are given the recursive rule:

$$x_{n+1} = \frac{x_n + p}{x_n + q}$$

We are also given $x_1 = 3$, $x_2 = 5$, and $x_3 = 7$. Using these values, we solve for $p$ and $q$:

$p = -33 \quad \text{and} \quad q = -9$.

Finally, we compute $x_4$ using the recursive formula:

$$x_4 = \frac{7 + (-33)}{7 + (-9)} = \frac{-26}{-2} = 13$$

Thus, the value of $x_4$ is 13.