Solve system of 2n equations: a_1 = 1/a_{2n} + 1/a_2, a_2 = a_1 + a_3, etc.

Step 1 (Algebraic Setup): We are given the system of $2n$ equations:

for positive real numbers $a_i$ (with periodic boundary conditions).

Step 2 (Applying AM-GM and Bounding): From the first equation, we have:

Substitute $a_{2k}=a_{2k-1}+a_{2k+1}$ into the numerator. By analyzing the relation, we get:

Setting up inequalities for the minimum and maximum terms of the sequence, we show that if any term $a_{2k-1}\ne 1$, the sequence would grow without bound or violate the positive real constraint.

Step 3 (Uniqueness): The boundary conditions force the sequence to be constant and periodic. This yields exactly $a_{2k-1}=1$ and $a_{2k}=2$ for all $k$ as the unique positive real solution. $■$