Let a,b be two real numbers between \(3\) and \(81 \)such that the resulting sequence \(3,a,b,81\) is in a geometric progression. The value of \(a+b\) is
Let a,b be two real numbers between \(3\) and \(81 \)such that the resulting sequence \(3,a,b,81\) is in a geometric progression. The value of \(a+b\) is
\(36\)
\(29\)
\(90\)
\(27\)
\(81\)
The Correct Option is A
Solution and Explanation
Given that
The G.P series is: \(3,a,b,81\)
means here first term is \(=3\)
last term \(=81\)
So, let
\(a=3.r\)
\( b=3.r=3r^2\)
Similarly, \(81=3r^3\)\(\)
\(⇒ 27=r^3\)\(\)
\(⇒ r=3\)
Therefore, \(a=3×3=9\)
\( b=3×3^2=27\)
So, \(a+b=9+27=36\) (_Ans)
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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