Question

Let a₁, a₂, a₃,.....,a_n 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

• A. 1
• B. n
• C. ⁿC₂
• D. 2

Answer 1

The Correct Option is B

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