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\[ 1 + 2 + \cdots + n = \frac{1}{2}n\left( {n + 1} \right) \] \[ 1^2 + 2^2 + 3^2 + \cdots + n^2 = \frac{1}{6}n\left( {n + 1} \right)\left( {2n + 1} \right) \] \[ 1^3 + 2^3 + 3^3 + \cdots + n^3 = \frac{1}{4}n^2 \left( {n + 1} \right)^2 \]
\[ 1^4 + 2^4 + 3^4 + \cdots + n^4 \] \[= \frac{1}{{30}}n\left( {n + 1} \right)\left( {2n + 1} \right)\left( {3n^2 + 3n - 1} \right) \]
\[ 1^5 + 2^5 + 3^5 + \cdots + n^5 \] \[= \frac{1}{{12}}n^2 \left( {n + 1} \right)^2 \left( {2n^2 + 2n - 1} \right) \]
\[ 1^6 + 2^6 + 3^6 + \cdots + n^6 \] \[= \frac{1}{{42}}n\left( {n + 1} \right)\left( {2n + 1} \right)\left( {3n^4 + 6n^3 - 3n + 1} \right) \]
\[ 1^7 + 2^7 + 3^7 + \cdots + n^7 \] \[= \frac{1}{{24}}n^2 \left( {n + 1} \right)^2 \left( {3n^4 + 6n^3 - n^2 - 4n + 2} \right) \]