First slide

Theory of expressions

Question

Let p, q be integers and let $α, β$ be the roots of $x^{2} - x - 1 = 0,$ where $α \neq β .$ For $n = 0, 1, 2, \dots$ Let $a_{n} = p α^{n} + q β^{2}$
Then,

Moderate

A

$a_{n + 1} = a_{n} + a_{n - 1}$
B

$a_{n + 2} = a_{n + 1} + a_{n - 1}$
C

$a_{n + 1} = a_{n} + 1$
D

$a_{n + 1} = a_{n - 1} + 1$

Solution

It is given that $α$ and $β$ are roots of $x^{2} - x - 1 = 0$
$∴ α + β = 1$ and $α β = - 1$
We have $a_{n} = p α^{n} + q β^{n}$
$\begin{array}{lrl} ∴ a_{n + 1} = p α^{n + 1} + q β^{n + 1} \\ \Rightarrow a_{n + 1} = (p α^{n} + q β^{n}) (α + β) - α β (p α^{n - 1} + q β^{n - 1}) \\ \Rightarrow a_{n + 1} = a_{n} + a_{n - 1} [∵ α + β = 1, α β = - 1] \end{array}$

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