Use a suitable telescoping sum to find a simpler expression for the sum $1^4+2^4+...+n^4$ where $n\in \mathbb{N}$
Prove by mathematical induction
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Chances are you have seen something similar done for $1^3+2^3+\cdots+n^3$, or maybe for $1^2+2^2+\cdots+n^2$, and you could try to do similar steps for this one. – Gerry Myerson Mar 07 '13 at 02:33
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This answer applies the technique to the sum of cubes; try using it as a model. – Brian M. Scott Mar 07 '13 at 02:36
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The suitable telescoping sum might be $$\sum_{k=1}^n \left[(k+1)^5-k^5\right]=(n+1)^5-1.$$ By expanding the left hand side and recalling formulas for $\sum k$, $\sum k^2$ and $\sum k^3$, you can solve for the desired sum.
Jonathan
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