Question

Sa se verifice egalitatea : [√1×2]+[√2×3]+...+[√50×51]=1275

Answer 1

   
[tex]\displaystyle \\ \text{Verificam daca:} \\ \sqrt{1\times 2} + \sqrt{2\times 3} + \sqrt{3\times 4} +\cdots+ \sqrt{50\times 51} = 1275 \\ \\ \sqrt{1\times 2} \ \textgreater \ \sqrt{1\times 1} \\ \sqrt{2\times 3} \ \textgreater \ \sqrt{2\times 2} \\ \sqrt{3\times 4} \ \textgreater \ \sqrt{3\times 3} \\ \text{.....................................} \\ \sqrt{50\times 51} \ \textgreater \ \sqrt{50\times 50} [/tex]


[tex]\displaystyle \\ \Longrightarrow~~\sqrt{1\times 2} + \sqrt{2\times 3} + \sqrt{3\times 4} +\cdots+ \sqrt{50\times 51} \ \textgreater \ \\ \ \textgreater \ \sqrt{1\times 1} + \sqrt{2\times 2} + \sqrt{3\times 3} +\cdots+ \sqrt{50\times 50} \\ \\ \sqrt{1\times 2} + \sqrt{2\times 3} + \sqrt{3\times 4} +\cdots+ \sqrt{50\times 51} \ \textgreater \ 1+ 2+ 3 +\cdots+ 50 \\ \\ \sqrt{1\times 2} + \sqrt{2\times 3} + \sqrt{3\times 4} +\cdots+ \sqrt{50\times 51} \ \textgreater \ \frac{50(50+1)}{2} [/tex]


[tex]\displaystyle \\ \sqrt{1\times 2} + \sqrt{2\times 3} + \sqrt{3\times 4} +\cdots+ \sqrt{50\times 51} \ \textgreater \ 25\times 51 \\ \\ \boxed{\sqrt{1\times 2} + \sqrt{2\times 3} + \sqrt{3\times 4} +\cdots+ \sqrt{50\times 51} \ \textgreater \ 1275 } [/tex]