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Math Forum » Discussions » Math Topics » alt.math.undergrad.independent

Topic: Induction proof
Replies: 24   Last Post: Aug 22, 2006 4:11 PM

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Dave L. Renfro

Posts: 4,448
Registered: 12/3/04
Re: Induction proof
Posted: Aug 19, 2006 5:55 PM
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kp wrote (in part):

> I want to understand why
> 1/(n+1)^2 < 1/(n*(n+1))

Statements:

1. 0 < 1

2. n < (n+1)

3. n(n+1) < (n+1)(n+1)

4. 1 / [n(n+1)] > 1 / [(n+1)(n+1)]

Reasons:

1. 1 is a positive number.

2. Add n to both sides of #1.

3. Multiply both sides of #2 by the positive number n+1.

4. Apply the function f(x) = 1/x, which is strictly decreasing
for x > 0, to both sides of #3. Recall that "f is strictly
decresing for x > 0" means "a < b and a,b > 0 ==> f(a) > f(b)"
is true.

Dave L. Renfro

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