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$\begin{array}{l} 2^2 + 6^2 + ... + \left( {4n - 2} \right)^2 + \left( {4n + 2} \right)^2 = \underbrace {2^2 + 6^2 + ... + \left( {4n - 2} \right)^2 }_{P\left( n \right)} + \left( {4n + 2} \right)^2 \\ . \\ =\frac{ 4n\left( {2n - 1} \right)\left( {2n + 1} \right)}{3} + \left( {4n + 2} \right)^2 \\ . \\ = \frac{4\left( {n + 1} \right)\left( {2n + 1} \right)\left( {2n + 3} \right)}{3} \\ \end{array}$

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$\rm{1+3+5+...+2n-1 este o suma in care termenii sunt nr.impare consecutive;\\2+4+6+...+2n este o suma in care termenii sunt nr. pare consecutive\\***\\1+2+3+...+100 este o suma la care, termenii sunt numere consecutive ...\\deci este de forma: \underbrace{1+2+3+...+n}_{suma lui Gauss}=\frac{n \cdot (n+1)}{2}\\3+7+11+15+...+43 nu este de o forma generala obisnuita ...\\ \Rightarrow trebuie sa gasim regula prin care aceasta suma a fost construita ...\\ " Observam ca termenii sumei cresc din 4 in 4 "\\Adica: \\3+7+11+15+...+43=3+(3+1*4)+(3+2*4)+(3+3*4)+...+(3+10*4)=\\=\underbrace{3+3+3+...+3}_{de 10+1-ori}+4*\underbrace{(1+2+3+...+10)}_{suma Gauss}=\\= ... s.a.m.$

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$\begin{array}{l} \sum\limits_{k = 1}^n {\left( {2k - 1} \right)} = 2 \cdot \sum\limits_{k = 1}^n k - \sum\limits_{k = 1}^n 1 \\ \sum\limits_{k = 1}^n k = \frac{{n(n + 1)}}{2} \\ \sum\limits_{k = 1}^n 1 = n \\ - - - - - - - - \\ \sum\limits_{k = 1}^n {\left( {2k - 1} \right)} = 2 \cdot \sum\limits_{k = 1}^n k - \sum\limits_{k = 1}^n 1 = 2 \cdot \frac{{n(n + 1)}}{2} - n = n^2 \\ {\rm Pt exercitiul tau,care cere sa se calculeze :} \\ {\rm 1 + 3 + 5 + }...{\rm + 99 = ??} \\ {\rm Putem fie folosim rezultatul de mai sus(mai scurt) fi ne folosim de observatia:} \\ {\rm 3 = 2 + 1} \\ {\rm 5 = 4 + 1} \\ {\rm 7 = 6 + 1} \\ .......... \\ {\rm 99 = 98 + 1} \\ {\rm Adunand toate sumele astea seobtine:} \\ {\rm 1 + 3 + }...{\rm + 99 = 1 + 2 + 1 + 4 + 1 + }....{\rm + 98 + 1} \\ {\rm = 50} \cdot {\rm 1 + 2 + }...{\rm + 98} \\ {\rm = 50 + 2(}\underbrace {{\rm 1 + }...{\rm + 49}}_{{\rm Suma}{\rm .Gauss}}{\rm )} \\ {\rm = 50 + 2}\left( {\frac{{{\rm 49} \cdot {\rm 50}}}{{\rm 2}}} \right) = ... = 50^2 \\ \end{array}$

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