Formulas of the sums of a number of natural numbers in integer degree

1 + 2 + 3 + 4 + 5 + .... + (n-1) + 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

 

1 ^ 2 + 2 ^ 2 + 3 ^ 2 + 4 ^ 2 + 5 ^ 2 + .... + (n-1) ^ 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

 

1 ^ 3 + 2 ^ 3 + 3 ^ 3 + 4 ^ 3 + 5 ^ 3 + .... + (n-1) ^ 3 + n ^ 3 = (\ frac {n (n + 1)} {2 }) ^ 2

 

The sum of the natural series where each element is raised to the fourth power

 

1 ^ 4 + 2 ^ 4 + 3 ^ 4 + 4 ^ 4 + 5 ^ 4 + .... + (n-1) ^ 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

 

1 ^ 5 + 2 ^ 5 + 3 ^ 5 + 4 ^ 5 + 5 ^ 5 + .... + (n-1) ^ 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

 

1 ^ 6 + 2 ^ 6 + 3 ^ 6 + 4 ^ 6 + 5 ^ 6 + .... + (n-1) ^ 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

 

1 ^ 7 + 2 ^ 7 + 3 ^ 7 + 4 ^ 7 + 5 ^ 7 + .... + (n-1) ^ 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

 

1 ^ 3 + 2 ^ 3 + 3 ^ 3 + 4 ^ 3 + 5 ^ 3 + .... + (n-1) ^ 3 + n ^ 3 = (\ frac {n (n-1)} {2 }) ^ 2

 

The sum of the natural series where each element is raised to the ninth degree

 

1 ^ 9 + 2 ^ 9 + 3 ^ 9 + 4 ^ 9 + 5 ^ 9 + .... + (n-1) ^ 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

1 ^ {10} + 2 ^ {10} + 3 ^ {10} + 4 ^ {10} + 5 ^ {10} + .... + (n-1) ^ {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

1 ^ {11} + 2 ^ {11} + 3 ^ {11} + 4 ^ {11} + 5 ^ {11} + .... + (n-1) ^ {11} + n ^ {11} = \ frac {1} {24} (2n ^ {12} + 12n ^ {11} + 22n ^ {10} -33n ^ 8 + 44n ^ 6-33n ^ 4 + 10n ^ 2)

 

The sum of the natural series where each element is raised to the twelfth degree

1 ^ {12} + 2 ^ {12} + 3 ^ {12} + 4 ^ {12} + 5 ^ {12} + .... + (n-1) ^ {12} + n ^ {12} = \ frac {1} {2730} (210n ^ {13} + 1365n ^ {12} + 2730n ^ {11} -5005n ^ 9 + 8580n ^ 7-9009n ^ 5 + 4550n ^ 3-691n)

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