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| Math Central | Quandaries & Queries |
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use mathematical induction to proof that each statement is true for every positve integer n 1/1.2+1/2.3+1/3.4+......1/n(n+1)=n/n+1 |
Samuel,
- Prove it's true for the first case:
1/(1*2) = 1/(1+1)
- Show that if it's true for the (n-1)st case it's true for the nth case [or that if it's true for the nth case it's true for the (n+1)st case, whichever is tidier.] To do this, use the fact that
[1/(1*2) + 1/(2*3) + ... + 1/n(n+1)] [*]
this is the nth left-hand side
= [1/(1*2) + 1/(2*3) + ... + 1/(n-1)n] + 1/n(n+1) [**]
this is the (n-1)st left-hand side
SUPPOSE [1/(1*2) + 1/(2*3) + ... + 1/(n-1)n] = (n-1)/n (the "inductive hypothesis") and substitute this into [**]. Simplify to get n/n+1; you have then shown that [*] = n/n+1 as desired.
-Good Hunting!
RD
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