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 statements is not true?

Moderate

A

$p_{3} = p_{5} - p_{4}$
B

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

$p_{5} = p_{2} p_{3}$
D

$p_{5^{'}} = 11$

Solution

It is given that $α$ and $β$ are roots of quadratic equation x² - x - 1=0, so sum of roots = $α$ + 0 = 1, and product of roots = $= α β = - 1$
and $p_{k} = α^{k} + β^{k}, k \geq 1$
so, $p_{1} = α + β = 1$
$p_{2} = α^{2} + β^{2} = (α + β)^{2} - 2 αβ = 1 + 2 = 3 p_{3} = α^{3} + β^{3} = (α + β)^{3} - 3 αβ (α + β) = 1 + 3 = 4 p_{4} = α^{4} + β^{4} = (α + β)^{4} - 4 αβ (α^{2} + β^{2}) - 6 α^{2} β^{2} = 1 + 12 - 6 = 7$
$and p_{5} = α^{5} + β^{5} = (α + β)^{5} - 5 αβ (α^{3} + β^{3}) - 10 α^{2} β^{2} (α + β) = 1 + 20 - 10 = 11 ∵ p_{3} = p_{5} - p_{4} = 4 p_{1} + p_{2} + p_{3} + p_{4} + p_{5} = 1 + 3 + 4 + 7 + 11 = 26 but p_{5} \neq p_{2} \cdot p_{3}$

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