If a1 (> 0), a2, a3, a4, a5 are in a G.P., a2 + a4 = 2a3 + 1 and 3a2 + a3 = 2a4, then a2 + a4 + 2a5 is equal to _______.
Correct Answer: 40
Solution and Explanation
The correct answer is 40
Let G.P. be a1 = a, a2 = ar, a3 = ar2, ……
\(∵ 3a_2 + a_3 = 2a_4\)
\(⇒ 3ar + ar^2 = 2ar³\)
⇒ 2ar² - r - 3 = 0
∴ r = -1 or \(\frac{3}{2}\)
∵ a1 = a > 0 then r ≠ -1
Now,
\(a_2 + a_4 = 2a_3 + 1\)
\(ar + ar³ = 2ar² + 1\)
\(a ( \frac{3}{2} + \frac{27}{8} - \frac{9}{2} ) = 1\)
\(∴ a = \frac{8}{3}\)
\(∴ \frac{8}{3} ( \frac{3}{2} + \frac{27}{8} + \frac{81}{8} )\)
= 40
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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