Calculus Examples

$\sum_{y = 0}^{8} - \frac{21 y}{5}$

Step 1

Split the summation to make the starting value of $y$ equal to $1$ .

$\sum_{y = 0}^{8} - \frac{21 y}{5} = \sum_{y = 1}^{8} - \frac{21 y}{5} + \sum_{y = 0}^{0} - \frac{21 y}{5}$

Step 2

Evaluate $\sum_{y = 1}^{8} - \frac{21 y}{5}$ .

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Step 2.1

Factor $- \frac{21}{5}$ out of the summation.

$(- \frac{21}{5}) \sum_{y = 1}^{8} y$

Step 2.2

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

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

Step 2.3

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

$(- \frac{21}{5}) (\frac{8 (8 + 1)}{2})$

Step 2.4

Simplify.

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Step 2.4.1

Add $8$ and $1$ .

$- \frac{21}{5} \cdot \frac{8 \cdot 9}{2}$

Step 2.4.2

Multiply $8$ by $9$ .

$- \frac{21}{5} \cdot \frac{72}{2}$

Step 2.4.3

Divide $72$ by $2$ .

$- \frac{21}{5} \cdot 36$

Step 2.4.4

Multiply $- \frac{21}{5} \cdot 36$ .

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Step 2.4.4.1

Multiply $36$ by $- 1$ .

$- 36 (\frac{21}{5})$

Step 2.4.4.2

Combine $- 36$ and $\frac{21}{5}$ .

$\frac{- 36 \cdot 21}{5}$

Step 2.4.4.3

Multiply $- 36$ by $21$ .

$\frac{- 756}{5}$

Step 2.4.5

Move the negative in front of the fraction.

$- \frac{756}{5}$

Step 3

Evaluate $\sum_{y = 0}^{0} - \frac{21 y}{5}$ .

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Step 3.1

Expand the series for each value of $y$ .

$- \frac{21 \cdot 0}{5}$

Step 3.2

Simplify.

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Step 3.2.1

Cancel the common factor of $0$ and $5$ .

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Step 3.2.1.1

Factor $5$ out of $21 \cdot 0$ .

$- \frac{5 (21 \cdot 0)}{5}$

Step 3.2.1.2

Cancel the common factors.

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Step 3.2.1.2.1

Factor $5$ out of $5$ .

$- \frac{5 (21 \cdot 0)}{5 (1)}$

Step 3.2.1.2.2

Cancel the common factor.

$- \frac{5 (21 \cdot 0)}{5 \cdot 1}$

Step 3.2.1.2.3

Rewrite the expression.

$- \frac{21 \cdot 0}{1}$

Step 3.2.1.2.4

Divide $21 \cdot 0$ by $1$ .

$- (21 \cdot 0)$

Step 3.2.2

Multiply $- (21 \cdot 0)$ .

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Step 3.2.2.1

Multiply $21$ by $0$ .

$- 0$

Step 3.2.2.2

Multiply $- 1$ by $0$ .

$0$

Step 4

Replace the summations with the values found.

$- \frac{756}{5} + 0$

Step 5

Add $- \frac{756}{5}$ and $0$ .

$- \frac{756}{5}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{756}{5}$

Decimal Form:

$- 151.2$

Mixed Number Form:

$- 151 \frac{1}{5}$