A geometric progression is a sequence in which each term is derived by multiplying or dividing the preceding term by a fixed number called the common ratio. E.g., the height to which a ball rises in each successive bounce follows a geometric progression. The sequence 4, -2, 1,... is a Geometric Progression (GP) for which (-1/2) is the common ratio. We can use the concept to find an arbitrary term, a finite or infinite sum of the series, and apply them in various contexts, including some difficult problems.
Suppose that the three distinct real numbers \(a,b \text{ and } c\) are in G.P. and \(a+b+c=xb\). Then
(A) \(-3<x<1 ;\)
(B) \(x>1\) or \(x<-3 ;\)
(C) \(x>3\) or \(x<-1 ;\)
(D) \(-1<x<3 ;\)
ISI entrance B. Stat. (Hons.) 2003 problem 3
Geometric Progression
6 out of 10
challenges and thrills of pre college mathematics
If any three quantity are in GP then we have a relation between them, in this case \(a,b,c\) are in G.P. so we have
\(b^2=ac\) or \(b= \sqrt{ac}\).
We also have \(a+b+c=xb\) so
which is equal to \(a+c=b(x-1)\)
=\(\frac{a}{b}+\frac{c}{b}=(x-1)\)
since \(b=\sqrt{ac}\) we will get
=\(\sqrt{\frac{a}{c}}+\sqrt{\frac{c}{a}}=(x-1)\)
=\(\sqrt{\frac{a}{c}}+\frac{1}{\sqrt{\frac{a}{c}}}=(x-1)\)
Let \(\sqrt{\frac{a}{c}}=k\), then we have the form of \(k+\frac{1}{k}\) which we know has a value either greater than 2 or less than -2.
so we can write
either, \(x-1 > 2\) or \(x-1 < -2\)
and now you can easily get the answer.
So the answer is x>3 or x<-1.
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