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posted by  Bob on 8/18/2008 5:23:42 PM  |  status: Live  

Discrete Math-Guess a formula and proove by induction

Course Textbook Chapter Problem
N/A Discrete Source ISBN 0536992908 1-7 25
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By experimenting with small values of n, guess a formula for the given sum


1/1.2 + 1/2.3 + ….+1/n(n+1) and use induction to prove the formula
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posted by zsm28 on 8/18/2008 5:40:34 PM  |  status: Live
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Response Details:
1/1.2 + 1/2.3 + 1/3.4
= 1/1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4
= 1/1 - 1/4
so guess
1/1.2 + 1/2.3 + ….+1/n(n+1) = 1 - 1/(n+1) = n/(n+1)
Now prove 1/1.2 + 1/2.3 + ….+1/n(n+1) = n/(n+1) using induction

For n = 1,
1/1.2 = 1/(1+1) = 1/2
it is true.

Assume it is true for n = k, that is
1/1.2 + 1/2.3 + ….+1/k(k+1) = k/(k+1)
Now for n = k + 1,
1/1.2 + 1/2.3 + ….+1/k(k+1) + 1/[(k+1)(k+2)]
= k/(k+1) + 1/[(k+1)(k+2)]
= 1 - 1/(k+1) + 1/(k+1) - 1/(k+2)
= 1 - 1/(k+2)
= (k+1)/[(k+1) + 1]
so it is also true for n = k +1.

By mathematical induction, the equality
1/1.2 + 1/2.3 + ….+1/n(n+1) = n/(n+1)
is true for any positive integer n.
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