Question:
If $a, b, c$ are in G.P., then
If $a, b, c$ are in G.P., then
Updated On: Jan 30, 2025
- $a^2, b^2, c^2$ are in G.P.
- $a^2 (b + c),c^2 (a + b),b^2 (a + c)$ are in G.P.
- $\frac{a}{b+c} , \frac{b}{c+a}, \frac{c}{a+b} $ are in G.P.
- None of these
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The Correct Option is A
Solution and Explanation
$\because a, b, c$ are in G.P.
$\therefore \frac{b}{a} = \frac{c}{b} =r$
$ \Rightarrow \frac{b^{2}}{a^{2}} = \frac{c^{2}}{b^{2}} =r^{2}$
$ \Rightarrow \frac{b^{2}}{a^{2}} = \frac{c^{2}}{b^{2}} =r^{2} $
$\Rightarrow a^{2} , b^{2} , c^{2} $ are in G.P.
$\therefore \frac{b}{a} = \frac{c}{b} =r$
$ \Rightarrow \frac{b^{2}}{a^{2}} = \frac{c^{2}}{b^{2}} =r^{2}$
$ \Rightarrow \frac{b^{2}}{a^{2}} = \frac{c^{2}}{b^{2}} =r^{2} $
$\Rightarrow a^{2} , b^{2} , c^{2} $ are in G.P.
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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