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**godelproof** $\displaystyle 1^{2} + 2^{2}+...(k-1)^{3} +((k+1)-1)^{3} < \frac{1}{4}(k+1)^{4} < 1^{3} +2^{3}+...k^{3} + (k+1)^{3}$ is what you are trying to prove by induction, NOT something you already know to be true! You want to prove $\displaystyle 1^{2} + 2^{2}+...(k-1)^{3} < \frac{1}{4}k^{4} < 1^{3} +2^{3}+...k^{3}$ $\displaystyle \Longrightarrow$ $\displaystyle 1^{2} + 2^{2}+...(k-1)^{3} +((k+1)-1)^{3} < \frac{1}{4}(k+1)^{4} < 1^{3} +2^{3}+...k^{3} + (k+1)^{3}$. How does it go from there?