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Q.

Let $α and β$ be the roots of the equation $x^{2} - x - 1 = 0.$ If $P_{k} = α^{k} + β^{k},$ for each $k \in N$ , then which one of the following statements is not true?

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a

$P_{3} = P_{5} - P_{4}$

b

$P_{5} = 11$

c

$P_{1} + P_{2} + P_{3} + P_{4} + P_{5} = 26$

d

$P_{5} = P_{2} . P_{3}$

answer is D.

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Detailed Solution

$P_{1} = α + β = 1, P_{2} = α^{2} + β^{2} = 1^{2} - 2 (- 1) = 3$

$P_{k + 2} - P_{k + 1} - P_{k} = 0$

$∴ P_{3} = P_{1} + P_{2} = 4, P_{4} = P_{2} + P_{3} = 7$

$P_{5} = P_{3} + P_{4} = 11$

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Let α and β be the roots of the equation x2−x−1=0. If Pk=αk+βk, for each k∈N, then which one of the following statements is not true?