orl wrote:
For every integer $ n \geq 2$ determine the minimum value that the sum $ \sum^n_{i = 0} a_i$ can take for nonnegative numbers $ a_0, a_1, \ldots, a_n$ satisfying the condition $ a_0 = 1,$ $ a_i \leq a_{i + 1} + a_{i + 2}$ for $ i = 0, \ldots, n - 2.$
It is easy understand that minimum value of the $ \sum^n_{i = 0} a_i$ will be then $ a_0, a_1, \ldots, a_n$ satisfying next algebraic system :
$ a_1 + a_2 = 1$
$ a_2 + a_3 = a_1$
$ a_3 + a_4 = a_2$
$ \ldots$
$ a_{k - 1} + a_{k} = a_{k - 2}$
$ \ldots$
$ a_{n - 1} + a_{n} = a_{n - 2}$
$ a_n = 0$
So: $ a_i = \frac {f_{n - i}}{f_n}$ - then $ f_i$ for $ i = 0, \ldots, n$ - Fibonacci series $ (f_0 = 0 ,f_1 = 1,f_2 = 1 , ...)$
min $ \sum^n_{i = 0} a_i = \frac {1}{f_n}$ $ (f_0 + f_1 + \ldots + f_{n - 1} + f_{n}) = 1 + \frac {f_{n + 1} - 1}{f_n}$