# Thread: Find the largest and smallest

1. ## Find the largest and smallest

The sum of the non-negative real numbers$\displaystyle s_1, s_2,..., s_{2004}$ is 2 and
$\displaystyle s_1s_2+s_2s_3+ ...+ s_{2003} s_{2004}+ s_{2004}s_1=1$.
Find the largest and smallest possible values of
$\displaystyle S=s_1^2+s_2^2+...+s_{2004}^2$

2. Originally Posted by perash
The sum of the non-negative real numbers$\displaystyle s_1, s_2,..., s_{2004}$ is 2 and
$\displaystyle s_1s_2+s_2s_3+ ...+ s_{2003} s_{2004}+ s_{2004}s_1=1$.
Find the largest and smallest possible values of
$\displaystyle S=s_1^2+s_2^2+...+s_{2004}^2$
$\displaystyle \sum_{i=1}^{2400} s_i =2$

Now square:

$\displaystyle \left(\sum_{i=1}^{2400} s_i\right)^2 = \sum_{i=1}^{2400} s_i^2 +2\sum_{i \ne j} s_i s_j = \sum_{i=1}^{2400} s_i^2 +2 =4$

So take it from there.

RonL

3. Originally Posted by CaptainBlack
$\displaystyle \left(\sum_{i=1}^{2400} s_i\right)^2 = \sum_{i=1}^{2400} s_i^2 +\color{red}2\sum_{i \ne j} s_i s_j \color{black}= \sum_{i=1}^{2400} s_i^2 +\color{red}2\color{black} =4$
No. $\displaystyle \sum_{i \ne j}{s_i s_j}$ is not the same as $\displaystyle s_1s_2+s_2s_3+\ldots+ s_{2003} s_{2004}+ s_{2004}s_1$. The former contains terms like $\displaystyle s_2s_4$ that are not present in the latter. Rather,

$\displaystyle S\ =\ s_1^2+s_2^2+\ldots+s_{2004}^2$

$\displaystyle =\ (s_1+\ldots+s_{2004})^2-2\sum_{i \ne j}{s_i s_j}$

$\displaystyle \leq\ (s_1+\ldots+s_{2004})^2-2(s_1s_2+s_2s_3+\ldots+ s_{2003} s_{2004}+ s_{2004}s_1)$

$\displaystyle =\ 2^2-2\ =\ 2$

For the lower bound, the Cauchy–Schwarz inequality gives

$\displaystyle |s_1s_2+s_2s_3+\ldots+ s_{2003} s_{2004}+ s_{2004}s_1|\ \leq\ (s_1^2+s_2^2+\ldots+s_{2004}^2)^{\frac{1}{2}}(s_2^ 2+s_3^2+\ldots+s_{2004}^2+s_1^2)^{\frac{1}{2}}$

so $\displaystyle S\ \geq\ 1$.