# $\sum _{k=1}^{n}{k}^{4}$の計算式

$\sum _{k=1}^{n}{k}^{4}={1}^{4}+{2}^{4}+{3}^{4}+\cdots +{n}^{4}$$=\frac{1}{30}n\left(n+1\right)\left(2n+1\right)\left(3{n}^{2}+3n-1\right)$

## ■公式の導出

${\left(k+1\right)}^{5}-{k}^{5}$$=5{k}^{4}+10{k}^{3}+10{k}^{2}+5k+1$ に順に$k=1,2,3,\cdots ,n$ 代入し，下のように縦にそろえて加えると

 ${2}^{5}-{1}^{5}$ $=$ $5·{1}^{4}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+10·{1}^{3}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+10·{1}^{2}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+5·1\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$ $\text{\hspace{0.17em}}+1$ ${3}^{5}-{2}^{5}$ $=$ $5·{2}^{4}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+10·{2}^{3}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+10·{2}^{2}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+5·2$ $\text{\hspace{0.17em}}+1$ ${4}^{5}-{3}^{5}$ $=$ $5·{3}^{4}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+10·{3}^{3}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+10·{3}^{2}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+5·3$ $\text{\hspace{0.17em}}+1$ $\cdot$ $\cdot$ $\cdot$ $+\right)\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}$ ${\left(n+1\right)}^{5}-{n}^{5}$ $=$ $5·{n}^{4}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+10·{n}^{3}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+10·{n}^{2}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}+5·n$ $\text{\hspace{0.17em}}+1$ $\overline{{\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\left(n+1\right)}^{5}-{1}^{5}\text{\hspace{0.17em}\hspace{0.17em}}=5\sum _{k-1}^{n}{k}^{4}+10\sum _{k-1}^{n}{k}^{3}+10\sum _{k-1}^{n}{k}^{2}+\sum _{k-1}^{n}k+n}$

${\left(n+1\right)}^{5}-1$$=5\sum _{k=1}^{n}{k}^{4}+10{\left\{\frac{1}{2}n\left(n+1\right)\right\}}^{2}$$+10\left\{\frac{1}{6}n\left(n+1\right)\left(2n+1\right)\right\}$$+5\left\{\frac{1}{2}n\left(n+1\right)\right\}+n$

となり，この式を整理すると

$5\sum _{k=1}^{n}{k}^{4}$$=\left\{{\left(n+1\right)}^{5}-1\right\}$$-\left[10{\left\{\frac{1}{2}n\left(n+1\right)\right\}}^{2}+10\left\{\frac{1}{6}n\left(n+1\right)\left(2n+1\right)\right\}+5\left\{\frac{1}{2}n\left(n+1\right)\right\}+n\right]$

$5\sum _{k=1}^{n}{k}^{4}={\left(n+1\right)}^{5}-1-10{\left\{\frac{1}{2}n\left(n+1\right)\right\}}^{2}-10\left\{\frac{1}{6}n\left(n+1\right)\left(2n+1\right)\right\}$$-5\left\{\frac{1}{2}n\left(n+1\right)\right\}-n$

$\sum _{k=1}^{n}{k}^{4}=\frac{1}{5}\left[{\left(n+1\right)}^{5}-1-10{\left\{\frac{1}{2}n\left(n+1\right)\right\}}^{2}-10\left\{\frac{1}{6}n\left(n+1\right)\left(2n+1\right)\right\}\right$$-5\left\{\frac{1}{2}n\left(n+1\right)\right\}-n]$

$=\frac{1}{5}\left[{\left(n+1\right)}^{5}-10×\frac{1}{4}{n}^{2}{\left(n+1\right)}^{2}-\frac{5}{3}n\left(n+1\right)\left(2n+1\right)-\frac{5}{2}n\left(n+1\right)-n-1\right]$

$=\frac{1}{5}\left\{{\left(n+1\right)}^{5}-\frac{5}{2}{n}^{2}{\left(n+1\right)}^{2}-\frac{5}{3}n\left(n+1\right)\left(2n+1\right)-\frac{5}{2}n\left(n+1\right)-\left(n+1\right)\right\}$

$=\frac{1}{5}\left(n+1\right)\left\{{\left(n+1\right)}^{4}-\frac{5}{2}{n}^{2}\left(n+1\right)-\frac{5}{3}n\left(2n+1\right)-\frac{5}{2}n-1\right\}$

$=\frac{1}{5}\left(n+1\right)\left({n}^{4}+4{n}^{3}+6{n}^{2}+4n+1-\frac{5}{2}{n}^{3}-\frac{5}{2}{n}^{2}-\frac{10}{3}{n}^{2}-\frac{5}{3}n-\frac{5}{2}n-1\right)$

$=\frac{1}{5}\left(n+1\right)\left\{{n}^{4}+\left(4-\frac{5}{2}\right){n}^{3}+\left(6-\frac{5}{2}-\frac{10}{3}\right){n}^{2}+\left(4-\frac{5}{3}-\frac{5}{2}\right)n\right\}$

$=\frac{1}{5}\left(n+1\right)\left\{{n}^{4}+\left(\frac{8-5}{2}\right){n}^{3}+\left(\frac{36-15-20}{6}\right){n}^{2}+\left(\frac{24-10-15}{6}\right)n\right\}$

$=\frac{1}{5}\left(n+1\right)\left({n}^{4}+\frac{3}{2}{n}^{3}+\frac{1}{6}{n}^{2}-\frac{1}{6}n\right)$

$=\frac{1}{30}n\left(n+1\right)\left(6{n}^{3}+9{n}^{2}+n-1\right)$

$f\left(-\frac{1}{2}\right)$$=6{\left(-\frac{1}{2}\right)}^{3}+9{\left(-\frac{1}{2}\right)}^{2}+\left(-\frac{1}{2}\right)-1$

$=-\frac{6}{8}+\frac{9}{4}-\frac{1}{2}-1$

$=-\frac{3}{4}+\frac{9}{4}-\frac{2}{4}-\frac{4}{4}$

$=\frac{-3+9-2-4}{4}$

$=0$

よって， $\text{\hspace{0.17em}}n+\frac{1}{2}\text{\hspace{0.17em}}$因数にもつので

 $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}6{n}^{2}+6n-2$ $n+\frac{1}{2}\text{\hspace{0.17em}}$ $\overline{\right)\text{\hspace{0.17em}}6{n}^{3}+9{n}^{2}+\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}n-1}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}6{n}^{3}+3{n}^{2}$ $\overline{\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}6{n}^{2}+\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}n\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}6{n}^{2}+3n$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\overline{\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}-2n-1}$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}-2n-1$ $\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}\overline{\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}0}$

$f\left(n\right)$$=\left(n+\frac{1}{2}\right)\left(6{n}^{2}+6n-2\right)$

$=2\left(n+\frac{1}{2}\right)\left(3{n}^{2}+3n-1\right)$

$=\left(2n+1\right)\left(3{n}^{2}+3n-1\right)$

これを代入すると

$\sum _{k=1}^{n}{k}^{4}$$=\frac{1}{30}n\left(n+1\right)\left(2n+1\right)\left(3{n}^{2}+3n-1\right)$

となり，$\sum _{k=1}^{n}{k}^{4}$ が求まる.

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