Question

Stiind ca $1^{2}$ + $2^{2}$ + $3^{2}$ .... + $31^{2}$ = 10 416 , calculati suma S = 3+5+8+12+...+467

Answer 1

[tex]1^2+2^2+...+30^2=10416-961=9455\\ 3=1+2\\ 5=(1+2)+2\\ 8=(1+2+3)+2\\ t_n= \frac{n(n+1)}{2} +2=>t_n= \frac{n^2+n}{2} +2\\ 467=\frac{n(n+1)}{2} +2\\ n(n+1)=930=30\cdot 31=>n=30\\ S=\frac{1^2+1}{2} +2+\frac{2^2+2}{2} +2+\frac{3^2+3}{2} +2+...+\frac{30^2+30}{2} +2\\ S= \frac{1^2+2^2+3^2+...+30^2}{2} + \frac{1+2+3+...+30}{2} +2\cdot 30\\ S= \frac{9455}{2} + \frac{31\cdot 30}{4} +60\\ S= \frac{9455}{2} + \frac{31\cdot 15}{2} +60\\ S= \frac{9455}{2} + \frac{465}{2} +\frac{120}{2}\\ S= \frac{10040}{2} \\ S=5020[/tex]