Calculus Examples

$\sum_{k = 1}^{9} k^{3}$

Step 1

The formula for the summation of a polynomial with degree $3$ is:

$\sum_{k = 1}^{n} k^{3} = \frac{n^{2} {(n + 1)}^{2}}{4}$

Step 2

Substitute the values into the formula.

$\frac{9^{2} {(9 + 1)}^{2}}{4}$

Step 3

Simplify.

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Simplify the numerator.

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Add $9$ and $1$ .

$\frac{9^{2} \cdot 10^{2}}{4}$

Raise $9$ to the power of $2$ .

$\frac{81 \cdot 10^{2}}{4}$

Raise $10$ to the power of $2$ .

$\frac{81 \cdot 100}{4}$

Simplify the expression.

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Multiply $81$ by $100$ .

$\frac{8100}{4}$

Divide $8100$ by $4$ .

$2025$