Question

1/2*3+1/3*4+1/4*5+......+1/9*10=?

Answer 1

general term can be written as
summation of
$\frac{1}{n(n + 1)} = \frac{1}{n} - \frac{1}{n + 1} \\ \frac{1}{2 \times 3} = \frac{1}{2} - \frac{1}{3} \\ \frac{1}{3 \times 4} = \frac{1}{3} - \frac{1}{4} \\ \frac{1}{4 \times 5} = \frac{1}{4} - \frac{1}{5} \\ .... \\ .... \\ .... \\ .... \\... \\ \frac{1}{8 \times 9} = \frac{1}{8} - \frac{1}{9} \\ \frac{1}{9 \times 10} = \frac{1}{9} - \frac{1}{10} \\ \\ add \: all\: \: these \\ lhs \: is \: what \: asked \: in \: ques \\ and \: see \: terms \: are \: being \: \\ canceled \: in \: rhs \: except( \frac{1}{2} - \frac{1}{10} ) \\ an s\: = \frac{1}{2} - \frac{1}{10} = \frac{8}{20} = \frac{2}{5}$
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Answer 2

1/2*3=1/2-1/3
1/3*4=1/3-1/4
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in that manner 1/9*10=1/9-1/10
1/2-1/3+1/3-1/4+1/4.................-1/9+1/9-1/10
finally 1/2-1/10 remains at last by cancelling all the terms.
1/2-1/10=8/20=2/5