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Let $α$ and $β$ be the roots of the equation $x^{2} - x - 1 = 0.$ If $P_{k} = {(α)}^{k} + {(β)}^{k} . k \geq 1$ , then match the columns
Column – I Column – II
I The value of $[\frac{p_{1} + p_{2} + p_{3} + p_{4} + p_{5}}{P_{2} P_{3}}]$ , (where [.] denotes GIF)is P 2
II $\frac{P_{4} + P_{2}}{P_{1} + P_{3}} =$ Q 3
III $P_{2} P_{3} - P_{5}$ R 4
IV $\frac{P_{1} + P_{5}}{P_{3}} =$ S 1

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$I \to P; I I \to R; I I I \to Q; I V \to S$

$I \to P; I I \to Q; I I I \to R; I V \to S$

$I \to Q; I I \to R; I I I \to S; I V \to P$

$I \to P; I I \to P; I I I \to S; I V \to Q$

answer is D.

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Detailed Solution

$α + β = 1 & α β = - 1 P_{1} = 1, P_{2} = 3, P_{3} = 4, P_{4} = 7 & P_{5} = 11$

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