Question

1/1x2+1/2x3+1/3x4+.......+1/14x15

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Answer 1

\(A=\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{14.15}\)

\(A=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{14}-\frac{1}{15}\)

\(A=1-\frac{1}{15}\)

\(A=\frac{14}{ }15\)

Answer 2

=1-1/2+1/2-1/3+...+1/14-1/15

=1-1/15

=14/15

Answer 3

\(\frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{14.15}=\left(1-\frac{1}{2}\right)+\left(\frac{1}{2}-\frac{1}{3}\right)+...+\left(\frac{1}{14}-\frac{1}{15}\right).\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-...-\frac{1}{14}+\frac{1}{14}-\frac{1}{15}=1-\left(\frac{1}{2}-\frac{1}{2}\right)_{ }-...-\left(\frac{1}{14}-\frac{1}{14}\right)-\frac{1}{15}_{ }.\)

\(=1-\frac{1}{15}=\frac{14}{15}.\)

Answer 4

\(\frac{1}{1\times2}+\frac{1}{2\times3}+.....+\frac{1}{14\times15}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-.....+\frac{1}{14}-\frac{1}{15}\)

\(=1-\frac{1}{15}\)

\(=\frac{14}{15}\)

Answer 5

=( 1/1 - 1/2 ) + ( 1/2 - 1/3 ) + ( 1/3 - 1/4 ) + ...+(1/14 - 1/15)

= ( 1/1 + 1/2 + 1/3 + ...1/14 ) +...+ ( 1/2 + 1/3 + 1/4 +...+ 1/15 )

= 1/1 - 1/15 

= 14/15