Question:medium
Let \(\alpha\) and \(\beta\) be the roots of the equation
\[
2x^2-5x-1=0.
\]
For \(n\in\mathbb{N}\), let
\[
P_n=\alpha^n+\beta^n.
\]
Then the value of
\[
\frac{2P_{11}\,(2P_{10}-5P_9)}{P_8\,(5P_{10}+P_9)}
\]
is equal to:
Let \(\alpha\) and \(\beta\) be the roots of the equation
\[
2x^2-5x-1=0.
\]
For \(n\in\mathbb{N}\), let
\[
P_n=\alpha^n+\beta^n.
\]
Then the value of
\[
\frac{2P_{11}\,(2P_{10}-5P_9)}{P_8\,(5P_{10}+P_9)}
\]
is equal to:
Show Hint
For sequences defined by roots of a quadratic equation,
always derive the recurrence relation first — it simplifies high-power expressions dramatically.
Updated On: Mar 3, 2026
- \(-\dfrac{1}{2}\)
- \(\dfrac{1}{2}\)
- \(-1\)
- \(1\)
Show Solution
The Correct Option is C
Solution and Explanation
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