Sum of a natural number
+n=\frac{n(n+1)}{2})
This is a well-known formula, discovered by Gauss at the age of six.
The sum of the natural series where each element is raised to the second degree
^2+n^2=\frac{1}{6}(2n^3+3n^2+n))
The sum of the natural series where each element is raised to the third degree
^3+n^3=(\frac{n(n+1)}{2})^2)
The sum of the natural series where each element is raised to the fourth power
^4+n^4=\frac{1}{30}(6n^5+15n^4+10n^3-n))
The sum of the natural series where each element is raised to the fifth power
^5+n^5=\frac{1}{12}(2n^6+6n^5+5n^4-n^2))
The sum of the natural series where each element is raised to the sixth power
^6+n^6=\frac{1}{42}(6n^7+21n^6+21n^5-7n^3+n))
The sum of the natural series where each element is raised to the seventh degree
^7+n^7=\frac{1}{24}(3n^8+12n^7+14n^6-7n^4+2n^2))
The sum of the natural series where each element is raised to the eighth power
^8+n^8=\frac{1}{360}(40n^9+180n^8+240n^7-168n^5+80n^3-12n))
The sum of the natural series where each element is raised to the ninth degree
^9+n^9=\frac{1}{180}(18n^{10}+90n^9+135n^8-126n^6+90n^4-27n^2))
The sum of the natural series where each element is raised to the tenth power
^{10}+n^{10}=\frac{1}{66}(6n^{11}+33n^{10}+55n^9-66n^7+66n^5-33n^3+5n))
The sum of the natural series where each element is elevated to the eleventh degree
^{11}+n^{11}=\frac{1}{24}(2n^{12}+12n^{11}+22n^{10}-33n^8+44n^6-33n^4+10n^2))
^{12}+n^{12}=\frac{1}{2730}(210n^{13}+1365n^{12}+2730n^{11}-5005n^9+8580n^7-9009n^5+4550n^3-691n))