For a natural number \(n\), let \(α_n = 19n – 12n\). Then, the value of \(\frac {31α_9−α_{10}}{57α_8}\) is _____.
Correct Answer: 4
Solution and Explanation
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 \(4\).
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Concepts Used:
Geometric Progression
What is Geometric Sequence?
A geometric progression is the sequence, in which each term is varied by another by a common ratio. The next term of the sequence is produced when we multiply a constant to the previous term. It is represented by: a, ar1, ar2, ar3, ar4, and so on.
Properties of Geometric Progression (GP)
Important properties of GP are as follows:
- Three non-zero terms a, b, c are in GP if b2 = ac
- In a GP,
Three consecutive terms are as a/r, a, ar
Four consecutive terms are as a/r3, a/r, ar, ar3 - In a finite GP, the product of the terms equidistant from the beginning and the end term is the same that means, t1.tn = t2.tn-1 = t3.tn-2 = …..
- If each term of a GP is multiplied or divided by a non-zero constant, then the resulting sequence is also a GP with a common ratio
- The product and quotient of two GP’s is again a GP
- If each term of a GP is raised to power by the same non-zero quantity, the resultant sequence is also a GP.
If a1, a2, a3,… is a GP of positive terms then log a1, log a2, log a3,… is an AP (arithmetic progression) and vice versa