If `alpha`, `beta` are the roots of the equation `ax^(2)+bx+c=0` and `S_(n)=alpha^(n)+beta^(n)`, then `aS_(n+1)+bS_(n)+cS_(n-1)=(n ge 2)`

Question

If `alpha`, `beta` are the roots of the equation `ax^(2)+bx+c=0` and `S_(n)=alpha^(n)+beta^(n)`, then `aS_(n+1)+bS_(n)+cS_(n-1)=(n ge 2)`

If `alpha`, `beta` are the roots of the equation `ax^(2)+bx+c=0` and `S_(n)=alpha^(n)+beta^(n)`, then `aS_(n+1)+bS_(n)+cS_(n-1)=(n ge 2)`
A. `0`
B. `a+b+c`
C. `(a+b+c)n`
D. `n^(2)abc`

Monjur Alahi

5 views

2025-01-04 04:42

1 Answers

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Salim Ahmed Kawsar · Accepted Answer · 2023-02-05T15:29:48+06:00

Salim Ahmed Kawsar

Correct Answer - A
`(a)` `S_(n+1)=alpha^(n+1)+beta^(n+1)`
`=(alpha+beta)(alpha^(n)+beta^(n))-alphabeta(alpha^(n-1)+beta^(n-1))`
`=-(b)/(a)*S_(n)-(c )/(a)*S_(n-1)`

5 views Answered 2023-02-05 15:29