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General sum

Can anyone tell me how to calculate general sum of sequence till \(n\) . \[1^{4}+2^{4}+3^{4}+ \ldots +n^{4}\]

Note by Akshat Sharda
2 years, 2 months ago

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Hint:

\[ \begin{eqnarray} \sum_{k=1}^n \left( (k+1)^2 - k^2 \right) &= &2 \sum_{k=1}^n k + \sum_{k=1}^n 1 \\ \displaystyle (n+1)^2 - 1 &=& 2 \color{blue}{\sum_{k=1}^n k} + n \\ \displaystyle \sum_{k=1}^n k &=& \frac{n(n+1)}2 \end{eqnarray} \]

And

\[ \begin{eqnarray} \sum_{k=1}^n \left( (k+1)^3 - k^3 \right) &= & \sum_{k=1}^n (3k^2 + 3k + 1) \\ \displaystyle (n+1)^3 - 1 &=& 3\color{blue}{ \sum_{k=1}^n k^2} +3 \sum_{k=1}^n k + \sum_{k=1}^n 1 \\ \displaystyle (n+1)^3 - 1 &=& 3\color{blue}{ \sum_{k=1}^n k^2} +3\frac{n(n+1)}2 + n \\ \sum_{k=1}^n k^2 &=& \frac16 n(n+1)(2n+1) \\ \end{eqnarray} \]

Can you find a general pattern to it?

Answer is \( \frac1{30} n(n+1)(2n+1)(3n^2 + 3n-1) \). You can double check your work by induction. Pi Han Goh · 2 years, 2 months ago

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@Pi Han Goh I appriciate what you did and Thank you for the same. Akshat Sharda · 2 years, 2 months ago

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@Pi Han Goh I have a question- Will this pattern work to find general sum of any power of \(n\)? Akshat Sharda · 2 years, 2 months ago

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@Akshat Sharda Study Bernoulli's Numbers. You'll get the answer for nth power sums. :) Satyajit Mohanty · 2 years, 2 months ago

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@Satyajit Mohanty I find that more troublesome, unless you got a table for the number of Bernoulli numbers, then yes, it will be easier, otherwise it takes up more time. Pi Han Goh · 2 years, 2 months ago

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@Pi Han Goh Well, it's true. I was just giving a general idea! :) Satyajit Mohanty · 2 years, 2 months ago

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@Akshat Sharda For positive integer powers, yes. Pi Han Goh · 2 years, 2 months ago

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Study Faulhaber's Formula for a General Idea. Also visit Bernoulli's numbers if it interests you. :) Satyajit Mohanty · 2 years, 2 months ago

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1/30(n+1)(2n+1)(3n^2+3n-1) Aakash Khandelwal · 2 years, 2 months ago

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(6x + 15x + 10x - x) / 30 Aashirvad Raj · 2 years, 2 months ago

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@Aashirvad Raj Hm, that simplifies to just \(x\). Are you missing some exponents? Calvin Lin Staff · 2 years, 2 months ago

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@Aashirvad Raj Even I know the formula but how to derive it is much more important. Akshat Sharda · 2 years, 2 months ago

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