Theory of Equations 1 Question 56
Passage Type Questions
Let $p, q$ be integers and let $\alpha, \beta$ be the roots of the equation, $x^{2}-x-1=0$ where $\alpha \neq \beta$. For $n=0,1,2, \ldots \ldots$, let $a _n=p \alpha^{n}+q \beta^{n}$.
FACT : If $a$ and $b$ are rational numbers and $a+b \sqrt{5}=0$, then $a=0=b$.
(2017 Adv.)
57. $a _{12}=$
(a) $a _{11}+2 a _{10}$
(b) $2 a _{11}+a _{10}$
(c) $a _{11}-a _{10}$
(d) $a _{11}+a _{10}$
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Answer:
Correct Answer: 57. (d)
Solution:
- $\quad \alpha^{2}=\alpha+1$
$ \beta^{2}=\beta+1 $
$a _n =p \alpha^{n}+q \beta^{n} $
$=p\left(\alpha^{n-1}+\alpha^{n-2}\right)+q\left(\beta^{n-1}+\beta^{n-2}\right) $
$=a _{n-1}+a _{n-2} $
$\therefore \quad a _{12} =a _{11}+a _{10}$
