Let a1, a2, a3,.....,an be positive real numbers.Then the minimum value of \(\frac{a_1}{a_2}+\frac{a_2}{a_3}+....+\frac{a_n}{a_1}\) is
- 1
- n
- nC2
- 2
The Correct Option is B
Solution and Explanation
Given positive real numbers a1,a2,...,a1,a2,...,an such that a1⋅a2⋅...⋅=a1⋅a2⋅...⋅an=c (equation 1), it is known that the arithmetic mean (A.M) is greater than or equal to the geometric mean (G.M), which can be expressed as:
1⋅2⋅...⋅−1⋅2≥(1⋅2⋅...⋅−1⋅(2))1nna1⋅a2⋅...⋅an−1⋅2an≥(a1⋅a2⋅...⋅an−1⋅(2an))n1
Simplifying:
(1+2+...+−1+2)≥2n(a1+a2+...+an−1+2an)≥nn2c
From equation (1), we know that 1+2+...+−1+2≥(2)1a1+a2+...+an−1+2an≥n(2c)n1.
Therefore, the minimum value of 1+2+...+−1+2a1+a2+...+an−1+2an is (2)1n(2c)n1.
The correct option is (B): n
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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