I need to show 1^3 + 2^3 + ... + n^3 < (1/2)n^4 for all n in N and n >=3.
I want to use mathematical induction, but I don't know if I need to use the first Mathematical induction or the second one?
Thanks
We are to prove that , for
The sum of cube series is equal to
We can restate the question as follows:
, for
Proof:
Base case: For integer ,
Induction hypothesis: Suppose for every integer
Then
LHS:
=
=
=
=
RHS:
=
Putting LHS and RHS together:
Hence, by induction hypothesis, , for all in
THanks!! it makes sense. My new question is that if what we are trying to prove the < (inequality), it seems that the RHS < LHS, because some terms are larger than others. However, there are other terms that are smaller than others. So, even though it seems logical that the inequality is true, is it enough to state it like that, or is there a need of more explanation to justify the inequality?