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How do you calculate 1/1*2*3 + 1/2*3*4... 1/98*99*100?

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parama9000 | Student, Grade 11 | (Level 1) Valedictorian

Posted September 14, 2013 at 1:07 PM via web

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How do you calculate 1/1*2*3 + 1/2*3*4... 1/98*99*100?

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llltkl | College Teacher | (Level 3) Valedictorian

Posted September 14, 2013 at 1:48 PM (Answer #1)

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Analyzing each term we get:

`1/(1*2*3)=1/2*(1/(1*2)-1/(2*3)) `
`1/(2*3*4)=1/2*(1/(2*3)-1/(3*4))`
`1/(3*4*5)=1/2*(1/(3*4)-1/(4*5))`
 

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`1/(98*99*100)=1/2*(1/(98*99)-1/(99*100))`

So, `1/(1*2*3)+1/(2*3*4)+1/(3*4*5)+....... 1/(98*99*100)`

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

`=1/2*(1/(1*2)-1/(99*100)) `

`=1/2*(1/2-1/9900)`

`=1/2*(4949/9900)`

`=4949/19800`    

Therefore, the answer is `4949/19800` .

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