1.

1^3 + 2^3 + ... + n3 = (n(n+1)/2 )^2 for any positive integer n.

2.

3^n < n! for all n > 6 with n ∈ N.

mathematical induction to prove these?

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- Nov 22nd 2008, 10:15 AMcaptainjapanProve
1.

1^3 + 2^3 + ... + n3 = (n(n+1)/2 )^2 for any positive integer n.

2.

3^n < n! for all n > 6 with n ∈ N.

mathematical induction to prove these? - Nov 22nd 2008, 10:32 AMo_O
Yes that's the way to go. Without going through all the formalities ...

#1: Assume it holds for . We want to show that it holds for as well, that is, we want to**show**that

So for your inductive step:

#2: Again, assume it holds for for all , so we're assuming that . We then want to**show**that it is true for as well, that is,

Note that:

We can assume the blue because we're given that the statement is true for values strictly greater than 6. So obviously, . Hopefully you'll see the conclusion directly after using the inductive step.