Series inequality induction proof

[Aristarchus_]

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...

 

[PeroK]

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.

 

[FactChecker]

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?

 

[pasmith]

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 $$1+2^{2} + 3^{2} + \dots +n^{2} > \frac {1}{3} n^{3}$$ I would use $$1 + 2^2 + 3^2 + \dots + n^2 = \tfrac16n(n+1)(2n+1).$$ 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.

 

[vela]

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?