Calculus Examples

$\sum_{k = 1}^{7} k^{3}$

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.

$\frac{7^{2} {(7 + 1)}^{2}}{4}$

Step 3

Simplify.

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Step 3.1

Simplify the numerator.

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Step 3.1.1

Add $7$ and $1$ .

$\frac{7^{2} \cdot 8^{2}}{4}$

Step 3.1.2

Raise $7$ to the power of $2$ .

$\frac{49 \cdot 8^{2}}{4}$

Step 3.1.3

Raise $8$ to the power of $2$ .

$\frac{49 \cdot 64}{4}$

Step 3.2

Simplify the expression.

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Step 3.2.1

Multiply $49$ by $64$ .

$\frac{3136}{4}$

Step 3.2.2

Divide $3136$ by $4$ .

$784$