Question:
The product of first nine terms of a GP is, in general, equal to which one of the following?
The product of first nine terms of a GP is, in general, equal to which one of the following?
Updated On: Jul 7, 2022
- The 9th power of the 4th term
- The 4th power of the 9th term
- The 5th power of the 9th term
- The 9th power of the 5th term
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The Correct Option is D
Solution and Explanation
Let a be the first term and r, the common ratio
First nine terms of a GP are $a, ar, ar^2, .... ar^8$.
$\therefore \, P = a.ar. ar^2 .... ar^8$
$= a^9.r^{1+2+...+8} $
$= a^9.r^{\frac{8.9}{2}}= a^9r^{36}$
$= (ar^4)^9 = (T_5)^9$
= 9th power of the 5th term
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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