Calculus Examples

$\sum_{k = 1}^{8} - 1 (\frac{2}{3}) k - 1$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $- 1 (\frac{2}{3})$ as $- (\frac{2}{3})$ .

$- (\frac{2}{3}) \cdot k - 1$

Step 1.1.2

Combine $k$ and $\frac{2}{3}$ .

$- \frac{k \cdot 2}{3} - 1$

Step 1.1.3

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

$- \frac{2 k}{3} - 1$

Step 1.2

Rewrite the summation.

$\sum_{k = 1}^{8} - \frac{2 k}{3} - 1$

Step 2

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

$\sum_{k = 1}^{8} - \frac{2 k}{3} - 1 = \sum_{k = 1}^{8} - \frac{2 k}{3} + \sum_{k = 1}^{8} - 1$

Step 3

Evaluate $\sum_{k = 1}^{8} - \frac{2 k}{3}$ .

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Step 3.1

Factor $- \frac{2}{3}$ out of the summation.

$(- \frac{2}{3}) \sum_{k = 1}^{8} k$

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.

$(- \frac{2}{3}) (\frac{8 (8 + 1)}{2})$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the expression.

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Step 3.4.1.1

Add $8$ and $1$ .

$- \frac{2}{3} \cdot \frac{8 \cdot 9}{2}$

Step 3.4.1.2

Multiply $8$ by $9$ .

$- \frac{2}{3} \cdot \frac{72}{2}$

Step 3.4.2

Cancel the common factor of $2$ .

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Step 3.4.2.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$\frac{- 2}{3} \cdot \frac{72}{2}$

Step 3.4.2.2

Factor $2$ out of $- 2$ .

$\frac{2 (- 1)}{3} \cdot \frac{72}{2}$

Step 3.4.2.3

Cancel the common factor.

$\frac{2 \cdot - 1}{3} \cdot \frac{72}{2}$

Step 3.4.2.4

Rewrite the expression.

$\frac{- 1}{3} \cdot 72$

Step 3.4.3

Cancel the common factor of $3$ .

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Step 3.4.3.1

Factor $3$ out of $72$ .

$\frac{- 1}{3} \cdot (3 (24))$

Step 3.4.3.2

Cancel the common factor.

$\frac{- 1}{3} \cdot (3 \cdot 24)$

Step 3.4.3.3

Rewrite the expression.

$- 1 \cdot 24$

Step 3.4.4

Multiply $- 1$ by $24$ .

$- 24$

Step 4

Evaluate $\sum_{k = 1}^{8} - 1$ .

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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.

$(- 1) (8)$

Step 4.3

Multiply $- 1$ by $8$ .

$- 8$

Step 5

Add the results of the summations.

$- 24 - 8$

Step 6

Subtract $8$ from $- 24$ .

$- 32$