Question

2012+\(\frac{2012}{1+2}\)+\(\frac{2012}{1+2+3}\) +\(\frac{2012}{1+2+3+4}\) +......+\(\frac{2012}{1+2+3+...+2011}\)

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Answer 1

\(2012+\frac{2012}{1+2}+\frac{2012}{1+2+3}+.....+\frac{2012}{1+2+3+....+2011}\)

\(=\frac{2012}{\frac{1\left(1+1\right)}{2}}+\frac{2012}{\frac{2\left(2+1\right)}{2}}+\frac{2012}{\frac{3\left(3+1\right)}{2}}+.....+\frac{2012}{\frac{2011\left(2011+1\right)}{2}}\)

\(=\frac{4024}{1.2}+\frac{4024}{2.3}+\frac{4024}{3.4}+.....+\frac{4024}{2011.2012}\)

\(=4024\left(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2011}-\frac{1}{2012}\right)\)

\(=4024\left(1-\frac{1}{2012}\right)\)

\(=4024.\frac{2011}{2012}\)

\(=4022\)