<s>[INST] How do I prove that $1^4+2^4+3^4\cdots\ + n^4 = \frac{1}{5}n^5 + \frac{1}{2}n^4 + \frac{1}{3}n^3-\frac{1}{30}n$? How do I prove that $1^4+2^4+3^4\cdots\ + n^4 = \frac{1}{5}n^5 + \frac{1}{2}n^4 + \frac{1}{3}n^3-\frac{1}{30}n$?
I've spent quite some time on this problem. So far, I've simplified the right-hand side to $\frac{1}{30}(n+1)[(2n+3)(3n^3)+n(n-1)]$. But then, the algebra becomes very complicated when I add $(n+1)^4$ to both sides of the inductive hypothesis.
 [/INST] The usual trick is to evaluate the sum $$\sum_{k=1}^n k(k+1)(k+2)(k+3)$$ using  $$k(k+1)(k+2)(k+3) = \dfrac{1}{5}k(k+1)(k+2)(k+3)(k+4) - \dfrac{1}{5}(k-1)k(k+1)(k+2)(k+3).$$ 
Then simply expand the summands to obtain $\sum_{k=1}^n k^4$.
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