Question

1/1x2+2/2x4+3/4x7+4/4x11+...+10/46x56

 

Answer 1

`[102]`

`= 1/1 - 1/2 + 1/2 - 1/4 + 1/4 - 1/7 + ... + 1/46 - 1/56`

`= 1/1 - 1/56`

`= 55/56`

Answer 2

\(\dfrac{1}{1\times2}+\dfrac{2}{2\times4}+...+\dfrac{10}{46\times56}\)

\(=\dfrac{2-1}{1\times2}+\dfrac{4-2}{2\times4}+...+\dfrac{56-46}{46\times56}\)

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+...+\dfrac{1}{46}-\dfrac{1}{56}\)

\(=1-\dfrac{1}{56}\)

\(=\dfrac{55}{56}\)

Answer 3

\(=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+...+\dfrac{1}{46}-\dfrac{1}{56}\)

=1-1/56=55/56

Answer 4

\(\dfrac{1}{1x2}+\dfrac{2}{2x4}+\dfrac{3}{4x7}+\dfrac{4}{7x11}+...+\dfrac{10}{46x56}\\ =\dfrac{2}{1x2}-\dfrac{1}{1x2}+\dfrac{4}{2x4}-\dfrac{2}{2x4}+\dfrac{7}{4x7}-\dfrac{4}{4x7}+\dfrac{11}{7x11}-\dfrac{7}{7x11}+...+\dfrac{56}{46x56}-\dfrac{46}{46x56}\\ =1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{11}+...+\dfrac{1}{46}-\dfrac{1}{56}\\ =1-\dfrac{1}{56}=\dfrac{55}{56}\)