First slide

Theory of equations

Question

$If α and β are the roots of the equation x^{2} - a x + b = 0 and A_{n} = α^{n} + β^{n}, then which of the following is true?$

Moderate

A

$A_{n + 1} = a A_{n} + b A_{n - 1}$
B

$A_{n + 1} = b A_{n} + a A_{n - 1}$
C

$A_{n + 1} = a A_{n} - b A_{n - 1}$
D

$A_{n + 1} = b A_{n} - a A_{n - 1}$

Solution

$\begin{array}{l} A_{n + 1} = α^{n + 1} + β^{n + 1} \\ a A_{n} - b A_{n - 1} = a (α^{n} + β^{n}) - b (α^{n - 1} + β^{n - 1}) \\ Now α + β = a, α β = b \end{array}$
$\begin{matrix} ∴ a A_{n} - b A_{n - 1} = (α + β) (α^{n} + β^{n}) - α β (α^{n - 1} + β^{n - 1}) \\ = α^{n + 1} + α β^{n} + α^{n} β + β^{n + 1} - α^{n} β - α β^{n} \\ = α^{n + 1} + β^{n + 1} = A_{n + 1} \end{matrix}$
$∴ A_{n + 1} = a A_{n} - b A_{n - 1}$

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