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}

 

Answers (1)

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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