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

Correct Answer: 4

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 \(4\).