MathDB

Let

n

be some positive integer and

a_1 , a_2 , \dots , a_n

be real numbers. Denote

S_0 = \sum_{i=1}^{n} a_i^2 , \hspace{1cm} S_1 = \sum_{i=1}^{n} a_ia_{i+1} , \hspace{1cm} S_2 = \sum_{i=1}^{n} a_ia_{i+2},

where

a_{n+1} = a_1

and

a_{n+2} = a_2.

1. Show that

S_0 - S_1 \geq 0

. 2. Show that

3

is the minimum value of

C

such that for any

n

and

a_1 , a_2 , \dots , a_n,

there holds

C(S_0 - S_1) \geq S_1 - S_2

A6