Question

For a natural number $n$, let $α_n = 19n – 12n$. Then, the value of $\frac {31α_9−α_{10}}{57α_8}$ is _____.

Answer 1

Given, $α_n = 19^n – 12^n$
Let equation of roots $12$ & $19$ i.e.
$x^2−31x+228=0$
Where, $x = 19$ or $12$
Thus,
$\frac {31α_9−α_{10}}{57α_8}=\frac {31(199−129)−(1910−1210)}{57(198−128)}$

$=\frac {199(31−19)−129(31−12)}{57(198−128)}$

$=\frac {228(198−128)}{57(198−128)}$

$= \frac {228}{57}$
$=4$

So, the answer is $4$.