Question

rezolvati ecuatia: -1+2+5+...+x=76​
Ajutor urgent!!!

Answer 1

 

$\displaystyle\bf\\-1+2+5+...+x=76\\\\Pastram~in~stanga~doar~termenii~pozitivi.\\\\2+5+...+x=76+1\\\\\boxed{\bf2+5+...+x=77}\\\\Calculam~numarul~de~termeni~in~functie~de~x.\\\\n=\frac{x-2}{3}+1=\frac{x-2}{3}+\frac{3}{3}=\frac{x-2+3}{3}\\\\\boxed{\bf n=\frac{x+1}{3}}$

.

$\displaystyle\bf\\Calculam~suma~termenilor~cu~formula~lui~Gauss.\\\\2+5+...+x=77\\\\\frac{n(x+2)}{2}=77\\\\\frac{\dfrac{x+1}{3}\cdot(x+2)}{2}=77\\\\\frac{(x+1)(x+2)}{6}=77\\\\ (x+1)(x+2)=77\cdot6\\\\ (x+1)(x+2)=462\\\\(x+1)~si~(x+2)~sunt~numere~consecutive.\\\\Descompunem~numarul~462~in~produs~de~2~numere~consecutive.\\\\462=21\cdot22\\\\\implies~x+1=21~si~x+2=22\\\\x=21-1\\\\\boxed{\bf x=20}$