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# $3 {\sum }\limits_{ k=1 }^{n} {k}^{4}$ の計算式

$3\displaystyle{ \displaystyle{{\sum }\limits_{ k=1 }^{n} {k}^{4} }={1}^{4}+{2}^{4}+{3}^{4}+\cdot \cdot \cdot +{n}^{4}=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 30 \hspace{2}}n \( n+1 \) \( 2n+1 \) \( 3{n}^{2}+3n- 1 \) }$

【式の導き方】
$3\displaystyle{{ \( k+1 \) }^{5}- {k}^{5}=5{k}^{4}+10{k}^{3}+10{k}^{2}+5k+1\text{\hspace{1}\hspace{1}}}$ に順に$3\displaystyle{k=1,2,3,\cdots ,n}$ 代入し，下のように縦にそろえて加えると，

 $3\displaystyle{{2}^{5}- {1}^{5}}$ $3\displaystyle{=}$ $3\displaystyle{5\cdot {1}^{4}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+10\cdot {1}^{3}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+10\cdot {1}^{2}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+5\cdot 1\text{\hspace{1}\hspace{1}\hspace{1}}}$ $3\displaystyle{\text{\hspace{1}}+1}$ $3\displaystyle{{3}^{5}- {2}^{5}}$ $3\displaystyle{=}$ $3\displaystyle{5\cdot {2}^{4}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+10\cdot {2}^{3}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+10\cdot {2}^{2}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+5\cdot 2}$ $3\displaystyle{\text{\hspace{1}}+1}$ $3\displaystyle{{4}^{5}- {3}^{5}}$ $3\displaystyle{=}$ $3\displaystyle{5\cdot {3}^{4}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+10\cdot {3}^{3}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+10\cdot {3}^{2}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+5\cdot 3}$ $3\displaystyle{\text{\hspace{1}}+1}$ $3\displaystyle{\cdot }$ $3\displaystyle{\cdot }$ $3\displaystyle{\cdot }$ $3\displaystyle{+\)\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}}$ $3\displaystyle{{ \( n+1 \) }^{5}- {n}^{5}}$ $3\displaystyle{=}$ $3\displaystyle{5\cdot {n}^{4}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+10\cdot {n}^{3}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+10\cdot {n}^{2}}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}}+5\cdot n}$ $3\displaystyle{\text{\hspace{1}}+1}$ $3\displaystyle{\bar{ { \text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}\( n+1 \) }^{5}- {1}^{5}\text{\hspace{1}\hspace{1}}=5\displaystyle{{\sum }\limits_{ k- 1 }^{n} {k}^{4} }+10\displaystyle{{\sum }\limits_{ k- 1 }^{n} {k}^{3} }+10\displaystyle{{\sum }\limits_{ k- 1 }^{n} {k}^{2} }+\displaystyle{{\sum }\limits_{ k- 1 }^{n}k}+n }}$

$3\displaystyle{{ \( n+1 \) }^{5}- 1=5\displaystyle{{\sum }\limits_{ k=1 }^{n} {k}^{4} }+10{ \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) \} }^{2}+10 \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}n \( n+1 \) \( 2n+1 \) \} +5 \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) \} +n}$

となり，この式を整理すると，

 $3\displaystyle{5\displaystyle{{\sum }\limits_{ k=1 }^{n} {k}^{4} }}$ $3\displaystyle{= \{ { \( n+1 \) }^{5}- 1 \} - \[ 10{ \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) \} }^{2}+10 \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}n \( n+1 \) \( 2n+1 \) \} +5 \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) \} +n \] }$ $3\displaystyle{5\displaystyle{{\sum }\limits_{ k=1 }^{n} {k}^{4} }}$ $3\displaystyle{={ \( n+1 \) }^{5}- 1- 10{ \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) \} }^{2}- 10 \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}n \( n+1 \) \( 2n+1 \) \} - 5 \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) \} - n}$ $3\displaystyle{\displaystyle{{\sum }\limits_{ k=1 }^{n} {k}^{4} }}$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}} \[ { \( n+1 \) }^{5}- 1- 10{ \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) \} }^{2}- 10 \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}n \( n+1 \) \( 2n+1 \) \} - 5 \{ \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) \} - n \] }$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}} \[ { \( n+1 \) }^{5}- 10\times \frac{\hspace{2}1\hspace{2}}{\hspace{2}4\hspace{2}}{n}^{2}{ \( n+1 \) }^{2}- \frac{\hspace{2}5\hspace{2}}{\hspace{2}3\hspace{2}}n \( n+1 \) \( 2n+1 \) - \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) - n- 1 \] }$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}} \{ { \( n+1 \) }^{5}- \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}{n}^{2}{ \( n+1 \) }^{2}- \frac{\hspace{2}5\hspace{2}}{\hspace{2}3\hspace{2}}n \( n+1 \) \( 2n+1 \) - \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}n \( n+1 \) - \( n+1 \) \} }$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}} \( n+1 \) \{ { \( n+1 \) }^{4}- \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}{n}^{2} \( n+1 \) - \frac{\hspace{2}5\hspace{2}}{\hspace{2}3\hspace{2}}n \( 2n+1 \) - \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}n- 1 \} }$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}} \( n+1 \) \( {n}^{4}+4{n}^{3}+6{n}^{2}+4n+1- \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}{n}^{3}- \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}{n}^{2}- \frac{\hspace{2} 10 \hspace{2}}{\hspace{2}3\hspace{2}}{n}^{2}- \frac{\hspace{2}5\hspace{2}}{\hspace{2}3\hspace{2}}n- \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}n- 1 \) }$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}} \( n+1 \) \{ {n}^{4}+ \( 4- \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}} \) {n}^{3}+ \( 6- \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}}- \frac{\hspace{2} 10 \hspace{2}}{\hspace{2}3\hspace{2}} \) {n}^{2}+ \( 4- \frac{\hspace{2}5\hspace{2}}{\hspace{2}3\hspace{2}}- \frac{\hspace{2}5\hspace{2}}{\hspace{2}2\hspace{2}} \) n \} }$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}} \( n+1 \) \{ {n}^{4}+ \( \frac{\hspace{2} 8- 5 \hspace{2}}{\hspace{2}2\hspace{2}} \) {n}^{3}+ \( \frac{\hspace{2} 36- 15- 20 \hspace{2}}{\hspace{2}6\hspace{2}} \) {n}^{2}+ \( \frac{\hspace{2} 24- 10- 15 \hspace{2}}{\hspace{2}6\hspace{2}} \) n \} }$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2}5\hspace{2}} \( n+1 \) \( {n}^{4}+\frac{\hspace{2}3\hspace{2}}{\hspace{2}2\hspace{2}}{n}^{3}+\frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}{n}^{2}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}6\hspace{2}}n \) }$ $3\displaystyle{=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 30 \hspace{2}}n \( n+1 \) \( 6{n}^{3}+9{n}^{2}+n- 1 \) }$

 $3\displaystyle{f \( - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \) }$ $3\displaystyle{=6{ \( - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \) }^{3}+9{ \( - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \) }^{2}+ \( - \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \) - 1}$ $3\displaystyle{=- \frac{\hspace{2}6\hspace{2}}{\hspace{2}8\hspace{2}}+\frac{\hspace{2}9\hspace{2}}{\hspace{2}4\hspace{2}}- \frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}- 1}$ $3\displaystyle{=- \frac{\hspace{2}3\hspace{2}}{\hspace{2}4\hspace{2}}+\frac{\hspace{2}9\hspace{2}}{\hspace{2}4\hspace{2}}- \frac{\hspace{2}2\hspace{2}}{\hspace{2}4\hspace{2}}- \frac{\hspace{2}4\hspace{2}}{\hspace{2}4\hspace{2}}}$ $3\displaystyle{=\frac{\hspace{2} - 3+9- 2- 4 \hspace{2}}{\hspace{2}4\hspace{2}}}$ $3\displaystyle{=0}$

よって， $3\displaystyle{\text{\hspace{1}}n+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\text{\hspace{1}}}$因数にもつので，

 $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}}6{n}^{2}+6n- 2}$ $3\displaystyle{n+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}}\text{\hspace{1}}}$ $3\displaystyle{\bar{ \)\text{\hspace{1}}6{n}^{3}+9{n}^{2}+\text{\hspace{1}\hspace{1}\hspace{1}}n- 1 }}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}}6{n}^{3}+3{n}^{2}}$ $3\displaystyle{\bar{ \text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}6{n}^{2}+\text{\hspace{1}\hspace{1}\hspace{1}}n\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}} }}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}6{n}^{2}+3n}$ $3\displaystyle{\bar{ \text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}- 2n- 1 }}$ $3\displaystyle{\text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}- 2n- 1}$ $3\displaystyle{\bar{ \text{\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}\hspace{1}}0 }}$

 $3\displaystyle{f \(n\) }$ $3\displaystyle{= \( n+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \) \( 6{n}^{2}+6n- 2 \) }$ $3\displaystyle{=2 \( n+\frac{\hspace{2}1\hspace{2}}{\hspace{2}2\hspace{2}} \) \( 3{n}^{2}+3n- 1 \) }$ $3\displaystyle{= \( 2n+1 \) \( 3{n}^{2}+3n- 1 \) }$

これを代入すると，

$3\displaystyle{\displaystyle{{\sum }\limits_{ k=1 }^{n} {k}^{4} }=\frac{\hspace{2}1\hspace{2}}{\hspace{2} 30 \hspace{2}}n \( n+1 \) \( 2n+1 \) \( 3{n}^{2}+3n- 1 \) }$

となり，$3\displaystyle{ {\sum }\limits_{ k=1 }^{n} {k}^{4} }$ が求まります.

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