Question

Aratati ca numarul a este natural de unde:a=(1/1×2+1/2×3+1/3×4+...+1/50×51)×(1+2+3+...+50)×2

Answer 1

$a=(\frac{1}{1\cdot 2}+\frac{1}{2\cdot 3}+\frac{1}{3\cdot 4}+...+\frac{1}{50\cdot 51})\cdot(1+2+3+...+50)\cdot 2\\</p><p>\\ a=(\frac{2-1}{1\cdot 2}+\frac{3-2}{2\cdot 3}+\frac{4-3}{3\cdot 4}+...+\frac{51-50}{50\cdot 51})\cdot(1+2+3+...+50)\cdot 2\\</p><p>\\ a=(\frac{2}{1\cdot2}-\frac{1}{1\cdot2}+\frac{3}{3\cdot 2}-\frac{2}{3\cdot 2}+\frac{4}{3\cdot 4}-\frac{3}{3\cdot 4}+...+\frac{51}{50\cdot51}-\frac{50}{50\cdot 51})\cdot (1+2+3+...+50)\cdot 2\\</p><p>\\ a=(\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{50}-\frac{1}{51})\cdot(1+2+3+...+50)\cdot 2\\</p><p>\\ a=(\frac{1}{1}-\frac{1}{51})\cdot \frac{50\cdot51}{\not{2}}\cdot \not{2}\\</p><p>\\ a=(\frac{51}{51}-\frac{1}{51})\cdot 50 \cdot 51\\</p><p>\\ a=\frac{51-1}{51}\cdot 50 \cdot 51\\</p><p>\\ a=50\cdot 50\\</p><p>\\ a=50^2=2500 \Rightarrow a~\in~N$