Quadratic Equation Question 3
Question 3 - 2024 (27 Jan Shift 2)
If $\alpha, \beta$ are the roots of the equation, $x^{2}-x-1=0$ and $S _n=2023 \alpha^{n}+2024 \beta^{n}$, then
(1) $2 S _{12}=S _{11}+S _{10}$
(2) $S _{12}=S _{11}+S _{10}$
(3) $2 S _{11}=S _{12}+S _{10}$
(4) $S _{11}=S _{10}+S _{12}$
Show Answer
Answer (2)
Solution
$x^{2}-x-1=0$
$S _n=2023 \alpha^{n}+2024 \beta^{n}$
$S _{n-1}+S _{n-2}=2023 \alpha^{n-1}+2024 \beta^{n-1}+2023 \alpha^{n-2}+2024 \beta^{n-2}$
$=2023 \alpha^{n-2}[1+\alpha]+2024 \beta^{n-2}[1+\beta]$
$=2023 \alpha^{n-2}\left[\alpha^{2}\right]+2024 \beta^{n-2}\left[\beta^{2}\right]$
$=2023 \alpha^{n}+2024 \beta^{n}$
$S _{n-1}+S _{n-2}=S _n$
Put $n=12$
$S _{11}+S _{10}=S _{12}$
