Thursday, November 29, 2018

Sum of k th power of positive integers (Determinant form)

$\newcommand{\D}{\displaystyle} \newcommand{\deta}[1]{\left|\begin{array}{cccccccc}#1\end{array}\right|}$

1. (Sum of the first $\D n$ positive integers)

 \[\sum_{i=1}^{n}i=\frac{1}{2!}\deta{(n+1)^2-(n+1)}\] 

2. (Sum of square of the first $\D n$ positive integers) 

\[\sum_{i=1}^{n}i^2=\frac{1}{3!}\deta{2&(n+1)^2-(n+1)\\3&(n+1)^3-(n+1)}\]

3. (Sum of cube of the first $\D n$ positive integers) 

\[\sum_{i=1}^{n}i^3=\frac{1}{4!}\deta{2&0&(n+1)^2-(n+1)\\3& 3& (n+1)^3-(n+1)\\4&6&(n+1)^4-(n+1)} \]

 4. (Sum of fouth power of the first $\D n$ positive integers) 

\[\sum_{i=1}^{n}i^4=\frac{1}{5!}\deta{2&0&0&(n+1)^2-(n+1)\\3& 3& 0&(n+1)^3-(n+1)\\4&6& 4&(n+1)^4-(n+1)\\ 5& 10&10& (n+1)^5-(n+1)} \]

 5. (Sum of $\D k^{th}$ power of the first $\D n$ positive integers) 

\[\sum_{i=1}^{n}i^k=\frac{1}{(k+1)!} \deta{ ^2C_1&0 & 0 &\cdots &\cdots &(n+1)^2-(n+1)\\ ^3C_1&^3C_2 & 0 &\cdots &\cdots &(n+1)^3-(n+1)\\ ^4C_1&^4C_2 & ^4C_3 &\cdots &\cdots &(n+1)^4-(n+1)\\ \vdots& \vdots& \vdots& \vdots& \vdots& \vdots \\ ^{k+1}C_1& ^{k+1}C_2& ^{k+1}C_3& \cdots& ^{k+1}C_{k-1}&(n+1)^{k+1}-(n+1) } \]

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