Let A1, A2, A3, … be an increasing geometric progression of positive real numbers. If A1A3A5A7 = \(\frac {1}{1296}\) and A2 + A4 = \(\frac {7}{36}\) then, the value of A6 + A8 + A10 is equal to
- 33
- 37
- 43
- 47
The Correct Option is C
Solution and Explanation
\(\frac {A_4}{r^3}. \frac {A_4}{r} . A_4r . A_4r^3 = \frac {1}{1296}\)
\(A_4 = \frac 16\)
\(A_2 = \frac {7}{36} - \frac 16\)
\(A_2= \frac {1}{36}\)
So, \(A_6 + A_8 + A_{10} = 1 + 6 + 36\)
\(= 43\)
So, the correct option is (C): \(43\)
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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