# If p,q are the roots of equation $a^{2}+bx+c=0$ and $S_{n}=p^{n}+q^{n}$ , then, $aS_{n+1}+cS_{n-1}=$ Option 1) $bS_{n}$ Option 2) $b^{2}S_{n}$ Option 3) $2bS_{n}$ Option 4) $-bS_{n}$

D Divya Saini

As we learnt in

Roots of Quadratic Equation with real Coefficients -

$\alpha ,\beta$ are roots if

$ax^{2}+bx+c= 0$

is satisfied by $x= \alpha ,\beta$

- wherein

$\alpha ,\beta\in C$

$a,b,c\in R$

$\\ax^{2}+bx+c=0\\*S_{n+1}=p^{n+1}+q^{n+1}\\*S_{n-1}=p^{n-1}+q^{n-1}\\*aS_{n+1}+cS_{n-1}\\*=\left ( ap^{2}+c \right )p^{n-1}+\left ( aq^{2}+c \right ) q^{n-1}\\\\*Now\, \, \, ap^{2}+bp+c=0\ \ and\ \ aq^{2}+bq+c=0\\* =-b\left ( p^{n} +q^{n}\right )= -bS_n$

Option 1)

$bS_{n}$

Incorrect

Option 2)

$b^{2}S_{n}$

Incorrect

Option 3)

$2bS_{n}$

Incorrect

Option 4)

$-bS_{n}$

Correct

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