If alpha ,beta are the roots of x^2 -ax + b = | vedclass

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Question

$\mathrm{x}^{2}-\mathrm{ax}+\mathrm{b}=0$ has roots $\alpha, \beta$

Then $\alpha+\beta=\mathrm{a}, \alpha \beta=\mathrm{b}, \mathrm{Now}$

$ \mat

Vedclass · Accepted Answer

$V_{n+1} = aV_n -bV_{n-1}$

Vedclass · Answer

$\mathrm{x}^{2}-\mathrm{ax}+\mathrm{b}=0$ has roots $\alpha, \beta$

Then $\alpha+\beta=\mathrm{a}, \alpha \beta=\mathrm{b}, \mathrm{Now}$

$ \mathrm{v}_{\mathrm{n}+1} =\alpha^{\mathrm{n}+1}+\beta^{\mathrm{n}+1} $

$=(\alpha+\beta)\left(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\right) $

$-\alpha \beta^{\mathrm{n}}-\beta \mathrm{a}^{\mathrm{n}} $

$=(\alpha+\beta)\left(\alpha^{\mathrm{n}}+\beta^{\mathrm{n}}\right)-\alpha \beta\left(\alpha^{\mathrm{n}-1}+\beta^{\mathrm{n}-1}\right) $

$ \mathrm{v}_{\mathrm{n}+1} =\mathrm{av}_{\mathrm{n}}-\mathrm{b} \mathrm{v}_{\mathrm{n}-1} $

If $\alpha ,\beta$ are the roots of $x^2 -ax + b = 0$ and if $\alpha^n + \beta^n = V_n$, then -

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