Question

Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:

• A. \( x^2 - x + 1 = 0 \)
• B. \( x^2 + x - 1 = 0 \)
• C. \( x^2 - x - 1 = 0 \)
• D. \( x^2 + x + 1 = 0 \)

Hint:

For such sequences, derive \( \alpha + \beta \) and \( \alpha \beta \) using known terms and build required equation.

Answer 1

The Correct Option is B

Given: \[ P_{10} = \alpha^{10} + \beta^{10} = 123, \quad P_9 = \alpha^9 + \beta^9 = 76, \quad P_8 = 47 \] From recurrence, determine: \[ \alpha + \beta = 1, \quad \alpha \beta = -1 \Rightarrow \text{Required equation with roots } \alpha \text{ and } \frac{1}{\beta} \Rightarrow x^2 + x - 1 = 0 \]