Question:
The sum of first 10 terms of a G.P. is equal to 244 times the sum of first five terms. Then the common ratio is
The sum of first 10 terms of a G.P. is equal to 244 times the sum of first five terms. Then the common ratio is
Updated On: Jul 23, 2024
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The Correct Option is B
Solution and Explanation
$S_{10} = 244 $
$ S_{5}\Rightarrow \frac{a\left(1-r^{10}\right)}{1-r} = 244 a\frac{\left(1-r^{5}\right)}{1-r} $
$ \Rightarrow 1-r^{10} = 244 \left(1-r^{5}\right) $
$ \Rightarrow \left(1-r^{5}\right)\left(1+r^{5}\right) = 244\left(1-r^{5}\right)$
$ \Rightarrow 1+r^{5}= 244$
$ \Rightarrow r^{5} = 243 = \left(3\right)^{5} $
$\Rightarrow r=3$
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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