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  4. For a natural number n, let a n =19 n -12 n . Then, the value of (31 α9-α10/57 α8) is

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Q. For a natural number $n$, let $a _{ n }=19^{ n }-12^{ n }$. Then, the value of $\frac{31 \alpha_{9}-\alpha_{10}}{57 \alpha_{8}}$ is

JEE MainJEE Main 2022Complex Numbers and Quadratic Equations

Solution:

$a _{ n }=19^{ n }-12^{ n }$
$\frac{31 \alpha_{9}-\alpha_{10}}{57 \alpha_{8}}=\frac{31\left(19^{9}-12^{9}\right)-\left(19^{10}-12^{10}\right)}{57 \alpha_{8}}$
$=\frac{19^{9}(31-19)-12^{9}(31-12)}{57 \alpha_{8}}$
$=\frac{19^{9} \cdot 12-12^{19} \cdot 19}{57 \alpha_{8}}$
$=\frac{12 \cdot 19\left(19^{8}-12^{8}\right)}{57 \alpha_{8}}=4$