Series inequality induction proof

Aristarchus_
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Homework Statement
How do I show that ##1+2^{2} + 3^{3} +...+n^{2} > \frac {1}{3} \cdot n^{3}## ?
Relevant Equations
k
My first attempt was ##... + n^{2} + (n+1)^{2} > \frac {1}{3} n^{3} + (n+1)^{2}##
then we must show that ##\frac {1}{3} n^{3} + (n+1)^{2} > \frac {1}{3} (n+1)^{3}##

We evaluate both sides and see that the LHS is indeed bigger than RHS. However, this solution is inconsistent so I am asking for some guidance as to a better method...
 
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What you have done does not look right as an attempt at an inductive proof.

Please make a genuine attempt at a proof by induction.
 
Is there a typo in your statement of the problem? In ##1+2^2+3^3+...+n^2##, I fail to see a pattern in the exponents. Is ##3^3## correct? Is ##n^2## correct?
 
Aristarchus_ said:
Homework Statement:: How do I show that ##1+2^{2} + 3^{3} +...+n^{2} > \frac {1}{3} \cdot n^{3}## ?
Relevant Equations:: k

My first attempt was ##... + n^{2} + (n+1)^{2} > \frac {1}{3} n^{3} + (n+1)^{2}##
then we must show that ##\frac {1}{3} n^{3} + (n+1)^{2} > \frac {1}{3} (n+1)^{3}##

We evaluate both sides and see that the LHS is indeed bigger than RHS. However, this solution is inconsistent so I am asking for some guidance as to a better method...

Assuming you mean <br /> 1+2^{2} + 3^{2} + \dots +n^{2} &gt; \frac {1}{3} n^{3}<br /> I would use <br /> 1 + 2^2 + 3^2 + \dots + n^2 = \tfrac16n(n+1)(2n+1). But if I was required to prove that result rather than just state it, then I would prefer your approach since it requires slightly less work.
 
Aristarchus_ said:
We evaluate both sides and see that the LHS is indeed bigger than RHS. However, this solution is inconsistent so I am asking for some guidance as to a better method...
What do you mean by the solution is inconsistent?
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...

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