Logical Sum of Fractions...

Number Theory Level 1

Find the sum of 1 1 × 2 + 1 2 × 3 + 1 3 × 4 + 1 4 × 5 + 1 5 × 6 + … + 1 99 × 100 \frac{1}{1\times 2 } + \frac{1}{ 2\times 3 } +\frac{ 1}{ 3\times 4} + \frac{1}{ 4\times 5} + \frac{1}{5\times 6}+\ldots+\frac{1}{99\times 100}


The answer is 0.99.

Satyen Nabar by Satyen Nabar , 48, India

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3 solutions

Hendrawan Tok
Apr 4, 2014

1/(1x2) = 1/1-1/2 ; 1/(2x3) = 1/2-1/3 ; 1/(3x4) = 1/3-1/4 and so on

1/1 -1/2+1/2-1/3+1/3-1/4+1/4 -1/5+1/5-1/6+...+1/98 -1/99+1/99 -1/100 = 1/1-1/100 = (100-1)/100

=99/100 = 0.99

2 Helpful 0 Interesting 0 Brilliant 0 Confused

The n t h \color{#20A900}{n^{th}} term can be expressed as 1 n ( n + 1 ) = 1 n − 1 n + 1 \color{#20A900}{\dfrac{1}{n(n+1)}=\dfrac{1}{n}-\dfrac{1}{n+1}} .So this is a telescoping series and it can be expressed as: 1 1 − 1 2 + 1 2 − 1 3 + ⋯ + 1 99 − 1 100 \color{#3D99F6}{\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\dotsm+\frac{1}{99}-\frac{1}{100}} All the intermediate terms cancel and we are left with 1 − 1 100 = 0.99 \color{#D61F06}{1-\frac{1}{100}=\boxed{0.99}}

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Satyen Nabar
Apr 4, 2014

1/2+1/6 = 3/6+1/6=4/6=2/3

2/3+ 1/12= 9/12=3/4

3/4+1/20= 16/20=4/5

Well that's the logical pattern

So answer will be 99/100 =0.99

1 Helpful 0 Interesting 0 Brilliant 0 Confused

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