Let alpha, beta be the roots of x^{2}-x-1=0 a | vedclass

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(A) Given that $\alpha$ and $\beta$ are the roots of $x^{2}-x-1=0$. Since $\alpha$ and $\beta$ are roots, they satisfy the equation: $\alpha^{2}-\alph

Vedclass · Accepted Answer

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

Vedclass · Answer

(A) Given that $\alpha$ and $\beta$ are the roots of $x^{2}-x-1=0$. Since $\alpha$ and $\beta$ are roots, they satisfy the equation: $\alpha^{2}-\alpha-1=0 \implies \alpha^{2}=\alpha+1$ $\beta^{2}-\beta-1=0 \implies \beta^{2}=\beta+1$ Multiplying by $\alpha^{n-1}$ and $\beta^{n-1}$ respectively: $\alpha^{n+1}=\alpha^{n}+\alpha^{n-1}$ $\beta^{n+1}=\beta^{n}+\beta^{n-1}$ Adding these two equations: $\alpha^{n+1}+\beta^{n+1}=(\alpha^{n}+\beta^{n})+(\alpha^{n-1}+\beta^{n-1})$ By definition of $S_{n}$, this is: $S_{n+1}=S_{n}+S_{n-1}$ Thus, option $A$ is correct.

Let $\alpha, \beta$ be the roots of $x^{2}-x-1=0$ and $S_{n}=\alpha^{n}+\beta^{n}$ for all integers $n \geq 1$. Then, for every integer $n \geq 2$, which of the following is true?

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