Question

If $\alpha$ and $\beta$ are the roots of the equation $x^2-ax+b=0$ and $A_n={\alpha }^n+{\beta }^n$, then which of the following is true?

• A. $A_{n+1}=a{A}_n-b{A}_{n-1}$
• B. $A_{n+1}=b{A}_n-a{A}_{n-1}$
• C. $A_{n+1}=b{A}_n+b{A}_{n-1}$
• D. $A_{n+1}=a{A}_n+b{A}_{n-1}$

Answer 1

The correct option is A $A_{n+1}=a{A}_n-b{A}_{n-1}$

$A_{n+1}={\alpha }^{n+1}+{\beta }^{n+1}$
$={\alpha }^{n+1}+{\alpha }^n\beta +{\beta }^{n+1}+\alpha {\beta }^n-{\alpha }^n\beta -\alpha {\beta }^n$
$={\alpha }^n(\alpha +\beta )+{\beta }^n(\beta +\alpha )-\alpha \beta ({\alpha }^{n-1}+{\beta }^{n-1})$
$=(\alpha +\beta )({\alpha }^n+{\beta }^n)-\alpha \beta ({\alpha }^{n-1}+{\beta }^{n-1})$
$=a{A}_n-b{A}_{n-1}$