New Formula for Powers of Natural Numbers
Posted on: April 26, 2025
In this post, I present a new pattern for expressing \(n^p\) in terms of sums of powers of integers from 2 to 6.
The following formulas express powers of n in terms of n and sums of powers:
\[ n^2 = n + 2\sum_{k=1}^{n-1} k \]
\[ n^3 = n + 3\left[\sum_{k=1}^{n-1} k + \sum_{k=1}^{n-1} k^2\right] \]
\[ n^4 = n + 4\left[\sum_{k=1}^{n-1} k + \frac{3}{2}\sum_{k=1}^{n-1} k^2 + \sum_{k=1}^{n-1} k^3\right] \]
\[ n^5 = n + 5\left[\sum_{k=1}^{n-1} k + \frac{1}{2}\sum_{k=1}^{n-1} k^2 + \frac{1}{2}\sum_{k=1}^{n-1} k^3 +
\sum_{k=1}^{n-1} k^4\right] \]
\[ n^6 = n + 6\left[\sum_{k=1}^{n-1} k + \frac{5}{2}\sum_{k=1}^{n-1} k^2 + \frac{10}{3}\sum_{k=1}^{n-1} k^3 +
\frac{5}{2}\sum_{k=1}^{n-1} k^4 + \sum_{k=1}^{n-1} k^5\right] \]
Notice that all formulas begin with n and then add a multiple of various sums. This reveals a fascinating relationship between powers of n and the sums of powers of integers less than n.