First slide

Theory of equations

Question

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 which one of the following statement is not true?

Moderate

A

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

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

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

$P_{5} = 11$

Solution

We have $α^{5} = 5 α + 3$ and $β^{5} = 5 β + 3$
$∴ P^{5} = α^{5} + β^{5} = 5 (α + β) + 6 = 5 (1) + 6 = 11$
$A l s o P_{2} = α^{2} + β^{2} = α + 1 + β + 1 = 3$
and $P_{3} = α^{3} + β^{3} = 2 α + 1 + 2 β + 1 = 2 (1) + 2 = 4$
$∴ P_{2} \times P_{3} = 12$ and $P_{5} = 11 \Rightarrow P_{5} \neq P_{2} \times P_{3}$

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