Question

1
. 1
1
Calculaţi: 1 -1
2 3 3 4
1 000​

Answer 1

 

$\displaystyle\bf\\1\frac{1}{2}\cdot1\frac{1}{3}\cdot1\frac{1}{4}\cdot1\frac{1}{5}\cdot...\cdot1\frac{1}{1000}=\\\\\\=1\frac{1}{2}\cdot1\frac{1}{3}\cdot1\frac{1}{4}\cdot1\frac{1}{5}\cdot...\cdot1\frac{1}{998}\cdot1\frac{1}{999}\cdot1\frac{1}{1000}=\\\\\\=\frac{1\cdot2+1}{2}\cdot\frac{1\cdot3+1}{3}\cdot\frac{1\cdot4+1}{4}\cdot\frac{1\cdot5+1}{5}\cdot...\\\\...\cdot\frac{1\cdot998+1}{998}\cdot\frac{1\cdot999+1}{999}\cdot\frac{1\cdot1000+1}{1000}=$

.

$\displaystyle\bf\\=\frac{3}{2}\cdot\frac{4}{3}\cdot\frac{5}{4}\cdot\frac{6}{5}\cdot...\cdot\frac{999}{998}\cdot\frac{1000}{999}\cdot\frac{1001}{1000}=\\\\Simplificam~numaratorul~fiecarei~fractii~cu~numitorul~fract~urmatoare.\\\\=\frac{1}{2}\cdot\frac{1}{1}\cdot\frac{1}{1}\cdot\frac{1}{1}\cdot...\cdot\frac{1}{1}\cdot\frac{1}{1}\cdot\frac{1001}{1}=\\\\\\=\frac{1}{2}\cdot\frac{1001}{1}=\boxed{\bf\frac{1001}{2}}=\boxed{\bf500\frac{1}{2}}= \boxed{\bf500,\!5}$