Calculus Examples

$\sum_{k = 1}^{30} k (k - 2) (k + 2)$

Step 1

Simplify the summation.

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Step 1.1

Apply the distributive property.

$(k \cdot k + k \cdot - 2) (k + 2)$

Step 1.2

Multiply $k$ by $k$ .

$(k^{2} + k \cdot - 2) (k + 2)$

Step 1.3

Move $- 2$ to the left of $k$ .

$(k^{2} - 2 \cdot k) (k + 2)$

Step 1.4

Expand $(k^{2} - 2 k) (k + 2)$ using the FOIL Method.

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Step 1.4.1

Apply the distributive property.

$k^{2} (k + 2) - 2 k (k + 2)$

Step 1.4.2

Apply the distributive property.

$k^{2} k + k^{2} \cdot 2 - 2 k (k + 2)$

Step 1.4.3

Apply the distributive property.

$k^{2} k + k^{2} \cdot 2 - 2 k \cdot k - 2 k \cdot 2$

Step 1.5

Simplify and combine like terms.

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Step 1.5.1

Simplify each term.

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Step 1.5.1.1

Multiply $k^{2}$ by $k$ by adding the exponents.

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Step 1.5.1.1.1

Multiply $k^{2}$ by $k$ .

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Step 1.5.1.1.1.1

Raise $k$ to the power of $1$ .

$k^{2} k^{1} + k^{2} \cdot 2 - 2 k \cdot k - 2 k \cdot 2$

Step 1.5.1.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$k^{2 + 1} + k^{2} \cdot 2 - 2 k \cdot k - 2 k \cdot 2$

Step 1.5.1.1.2

Add $2$ and $1$ .

$k^{3} + k^{2} \cdot 2 - 2 k \cdot k - 2 k \cdot 2$

Step 1.5.1.2

Move $2$ to the left of $k^{2}$ .

$k^{3} + 2 \cdot k^{2} - 2 k \cdot k - 2 k \cdot 2$

Step 1.5.1.3

Multiply $k$ by $k$ by adding the exponents.

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Step 1.5.1.3.1

Move $k$ .

$k^{3} + 2 k^{2} - 2 (k \cdot k) - 2 k \cdot 2$

Step 1.5.1.3.2

Multiply $k$ by $k$ .

$k^{3} + 2 k^{2} - 2 k^{2} - 2 k \cdot 2$

Step 1.5.1.4

Multiply $2$ by $- 2$ .

$k^{3} + 2 k^{2} - 2 k^{2} - 4 k$

Step 1.5.2

Subtract $2 k^{2}$ from $2 k^{2}$ .

$k^{3} + 0 - 4 k$

Step 1.5.3

Add $k^{3}$ and $0$ .

$k^{3} - 4 k$

Step 1.6

Rewrite the summation.

$\sum_{k = 1}^{30} k^{3} - 4 k$

Step 2

Split the summation into smaller summations that fit the summation rules.

$\sum_{k = 1}^{30} k^{3} - 4 k = \sum_{k = 1}^{30} k^{3} - 4 \sum_{k = 1}^{30} k$

Step 3

Evaluate $\sum_{k = 1}^{30} k^{3}$ .

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Step 3.1

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 3.2

Substitute the values into the formula.

$\frac{30^{2} {(30 + 1)}^{2}}{4}$

Step 3.3

Simplify.

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Step 3.3.1

Simplify the numerator.

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Step 3.3.1.1

Add $30$ and $1$ .

$\frac{30^{2} \cdot 31^{2}}{4}$

Step 3.3.1.2

Raise $30$ to the power of $2$ .

$\frac{900 \cdot 31^{2}}{4}$

Step 3.3.1.3

Raise $31$ to the power of $2$ .

$\frac{900 \cdot 961}{4}$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $900$ by $961$ .

$\frac{864900}{4}$

Step 3.3.2.2

Divide $864900$ by $4$ .

$216225$

Step 4

Evaluate $4 \sum_{k = 1}^{30} k$ .

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Step 4.1

The formula for the summation of a polynomial with degree $1$ is:

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

Step 4.2

Substitute the values into the formula and make sure to multiply by the front term.

$(- 4) (\frac{30 (30 + 1)}{2})$

Step 4.3

Simplify.

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Step 4.3.1

Simplify the expression.

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Step 4.3.1.1

Add $30$ and $1$ .

$- 4 \frac{30 \cdot 31}{2}$

Step 4.3.1.2

Multiply $30$ by $31$ .

$- 4 (\frac{930}{2})$

Step 4.3.2

Cancel the common factor of $2$ .

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Step 4.3.2.1

Factor $2$ out of $- 4$ .

$2 (- 2) \frac{930}{2}$

Step 4.3.2.2

Cancel the common factor.

$2 \cdot - 2 \frac{930}{2}$

Step 4.3.2.3

Rewrite the expression.

$- 2 \cdot 930$

Step 4.3.3

Multiply $- 2$ by $930$ .

$- 1860$

Step 5

Add the results of the summations.

$216225 - 1860$

Step 6

Subtract $1860$ from $216225$ .

$214365$