Calculus Examples

$\sum_{k = 1}^{30} (\frac{2}{15} k + 1) \frac{2}{15}$

Step 1

Simplify the summation.

Tap for more steps...

Step 1.1

Combine $\frac{2}{15}$ and $k$ .

$(\frac{2 k}{15} + 1) \frac{2}{15}$

Step 1.2

Apply the distributive property.

$\frac{2 k}{15} \cdot \frac{2}{15} + 1 (\frac{2}{15})$

Step 1.3

Multiply $\frac{2 k}{15} \cdot \frac{2}{15}$ .

Tap for more steps...

Step 1.3.1

Multiply $\frac{2 k}{15}$ by $\frac{2}{15}$ .

$\frac{2 k \cdot 2}{15 \cdot 15} + 1 (\frac{2}{15})$

Step 1.3.2

Multiply $2$ by $2$ .

$\frac{4 k}{15 \cdot 15} + 1 (\frac{2}{15})$

Step 1.3.3

Multiply $15$ by $15$ .

$\frac{4 k}{225} + 1 (\frac{2}{15})$

Step 1.4

Multiply $\frac{2}{15}$ by $1$ .

$\frac{4 k}{225} + \frac{2}{15}$

Step 1.5

Rewrite the summation.

$\sum_{k = 1}^{30} \frac{4 k}{225} + \frac{2}{15}$

Step 2

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

$\sum_{k = 1}^{30} \frac{4 k}{225} + \frac{2}{15} = \sum_{k = 1}^{30} \frac{4 k}{225} + \sum_{k = 1}^{30} \frac{2}{15}$

Step 3

Evaluate $\sum_{k = 1}^{30} \frac{4 k}{225}$ .

Tap for more steps...

Step 3.1

Factor $\frac{4}{225}$ out of the summation.

$(\frac{4}{225}) \sum_{k = 1}^{30} k$

Step 3.2

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

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

Step 3.3

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

$(\frac{4}{225}) (\frac{30 (30 + 1)}{2})$

Step 3.4

Simplify.

Tap for more steps...

Step 3.4.1

Simplify the expression.

Tap for more steps...

Step 3.4.1.1

Add $30$ and $1$ .

$\frac{4}{225} \cdot \frac{30 \cdot 31}{2}$

Step 3.4.1.2

Multiply $30$ by $31$ .

$\frac{4}{225} \cdot \frac{930}{2}$

Step 3.4.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 3.4.2.1

Factor $2$ out of $4$ .

$\frac{2 (2)}{225} \cdot \frac{930}{2}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 \cdot 2}{225} \cdot \frac{930}{2}$

Step 3.4.2.3

Rewrite the expression.

$\frac{2}{225} \cdot 930$

Step 3.4.3

Cancel the common factor of $15$ .

Tap for more steps...

Step 3.4.3.1

Factor $15$ out of $225$ .

$\frac{2}{15 (15)} \cdot 930$

Step 3.4.3.2

Factor $15$ out of $930$ .

$\frac{2}{15 \cdot 15} \cdot (15 \cdot 62)$

Step 3.4.3.3

Cancel the common factor.

$\frac{2}{15 \cdot 15} \cdot (15 \cdot 62)$

Step 3.4.3.4

Rewrite the expression.

$\frac{2}{15} \cdot 62$

Step 3.4.4

Combine $\frac{2}{15}$ and $62$ .

$\frac{2 \cdot 62}{15}$

Step 3.4.5

Multiply $2$ by $62$ .

$\frac{124}{15}$

Step 4

Evaluate $\sum_{k = 1}^{30} \frac{2}{15}$ .

Tap for more steps...

Step 4.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 4.2

Substitute the values into the formula.

$(\frac{2}{15}) (30)$

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Cancel the common factor of $15$ .

Tap for more steps...

Step 4.3.1.1

Factor $15$ out of $30$ .

$\frac{2}{15} (15 (2))$

Step 4.3.1.2

Cancel the common factor.

$\frac{2}{15} (15 \cdot 2)$

Step 4.3.1.3

Rewrite the expression.

$2 \cdot 2$

Step 4.3.2

Multiply $2$ by $2$ .

$4$

Step 5

Add the results of the summations.

$\frac{124}{15} + 4$

Step 6

Simplify.

Tap for more steps...

Step 6.1

To write $4$ as a fraction with a common denominator, multiply by $\frac{15}{15}$ .

$\frac{124}{15} + 4 \cdot \frac{15}{15}$

Step 6.2

Combine $4$ and $\frac{15}{15}$ .

$\frac{124}{15} + \frac{4 \cdot 15}{15}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{124 + 4 \cdot 15}{15}$

Step 6.4

Simplify the numerator.

Tap for more steps...

Step 6.4.1

Multiply $4$ by $15$ .

$\frac{124 + 60}{15}$

Step 6.4.2

Add $124$ and $60$ .

$\frac{184}{15}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{184}{15}$

Decimal Form:

$12.2\overline{6}$

Mixed Number Form:

$12 \frac{4}{15}$