Mathematics Question on geometric progression

Question

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

Answer 1

Solution

Given positive real numbers a1,a2,...,a 1,a 2,...,an such that a1⋅a2⋅...⋅=a 1⋅ a 2⋅...⋅ 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))1 nna 1⋅ a 2⋅...⋅ an −1⋅2 an ≥(a 1⋅ a 2⋅...⋅ an −1⋅(2 an ))n 1

Simplifying:

(1+2+...+−1+2)≥2 n(a 1+a 2+...+an −1+2 an )≥ nn 2 c

From equation (1), we know that 1+2+...+−1+2≥(2)1 a 1+a 2+...+an −1+2 an ≥ n(2 c)n 1.

Therefore, the minimum value of 1+2+...+−1+2 a 1+a 2+...+an −1+2 an is (2)1 n(2 c)n 1.

The correct option is (B): n