Question

Aratati ca S = 1 pe (2 la 2) + 1 pe( 3 la 2) + 1 pe (4 la 2) + ... + 1 pe (2011 la 2) este mai mic ca 2010 pe 2011​

Answer 1

$\mathbf{S=\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2} +...+\dfrac{1}{2011^2} <\dfrac{2010}{2011} }$

$\mathbf{\implies \dfrac{1}{2^2} <\dfrac{1}{1\cdot 2} }$

$\mathbf{\implies \dfrac{1}{3^2} <\dfrac{1}{2 \cdot 3} }$

$\mathbf{\implies \dfrac{1}{4^2} <\dfrac{1}{3 \cdot 4} }$

$\mathbf{\implies \dfrac{1}{2011^2} <\dfrac{1}{2010\cdot 2011} }$

$\mathbf{\implies S<S_2}$

$\mathbf{S_2=\dfrac{1}{1\cdot 2}+\dfrac{1}{2\cdot 3} +\dfrac{1}{3\cdot 4} +...+\dfrac{1}{2010\cdot 2011} }$

$\mathbf{S_2=\dfrac{2-1}{1\cdot 2}+\dfrac{3-2}{2\cdot 3} +\dfrac{4-3}{3\cdot 4}+...+\dfrac{2011-2010}{2010 \cdot 2011} }$

$\mathbf{S_2=\dfrac{1}{1}-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3} -\dfrac{1}{4} +...+\dfrac{1}{2010} -\dfrac{1}{2011} }$

$\mathbf{ Observam \: ca \: toate \: fractiile \:se \: simplifica\: cu \:exceptia\:primei \: si \: a \: ultimei}$

$\mathbf{\implies S_2= \dfrac{1}{1} -\dfrac{1}{2011}=\dfrac{2011}{2011}-\dfrac{1}{2011} =\dfrac{2010}{2011} }$

$\mathbf{Stiind \:\; ca \:\; S<S_2 \implies S<\dfrac{2010}{2011} }$

${\boxed{\mathbf{\implies \dfrac{1}{2^2}+\dfrac{1}{3^2} +\frac{1}{4^2}+...+\frac{1}{2011^2 } <\dfrac{2010}{2011} }}}$