Calculus Examples

$\sum_{k = 5}^{13} \frac{5 - 2 k}{8}$

Step 1

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

$\sum_{k = 5}^{13} \frac{5}{8} - \frac{k}{4} = \sum_{k = 1}^{13} \frac{5}{8} - \frac{k}{4} - \sum_{k = 1}^{4} \frac{5}{8} - \frac{k}{4}$

Step 2

Evaluate $\sum_{k = 1}^{13} \frac{5}{8} - \frac{k}{4}$ .

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Step 2.1

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

$\sum_{k = 1}^{13} \frac{5}{8} - \frac{k}{4} = \sum_{k = 1}^{13} \frac{5}{8} + \sum_{k = 1}^{13} - \frac{k}{4}$

Step 2.2

Evaluate $\sum_{k = 1}^{13} \frac{5}{8}$ .

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Step 2.2.1

The formula for the summation of a constant is:

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

Step 2.2.2

Substitute the values into the formula.

$(\frac{5}{8}) (13)$

Step 2.2.3

Multiply $(\frac{5}{8}) (13)$ .

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Step 2.2.3.1

Combine $\frac{5}{8}$ and $13$ .

$\frac{5 \cdot 13}{8}$

Step 2.2.3.2

Multiply $5$ by $13$ .

$\frac{65}{8}$

Step 2.3

Evaluate $\sum_{k = 1}^{13} - \frac{k}{4}$ .

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Step 2.3.1

Factor $- \frac{1}{4}$ out of the summation.

$(- \frac{1}{4}) \sum_{k = 1}^{13} k$

Step 2.3.2

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

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

Step 2.3.3

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

$(- \frac{1}{4}) (\frac{13 (13 + 1)}{2})$

Step 2.3.4

Simplify.

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Step 2.3.4.1

Simplify the expression.

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Step 2.3.4.1.1

Add $13$ and $1$ .

$- \frac{1}{4} \cdot \frac{13 \cdot 14}{2}$

Step 2.3.4.1.2

Multiply $13$ by $14$ .

$- \frac{1}{4} \cdot \frac{182}{2}$

Step 2.3.4.2

Cancel the common factor of $2$ .

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Step 2.3.4.2.1

Move the leading negative in $- \frac{1}{4}$ into the numerator.

$\frac{- 1}{4} \cdot \frac{182}{2}$

Step 2.3.4.2.2

Factor $2$ out of $4$ .

$\frac{- 1}{2 (2)} \cdot \frac{182}{2}$

Step 2.3.4.2.3

Factor $2$ out of $182$ .

$\frac{- 1}{2 \cdot 2} \cdot \frac{2 \cdot 91}{2}$

Step 2.3.4.2.4

Cancel the common factor.

$\frac{- 1}{2 \cdot 2} \cdot \frac{2 \cdot 91}{2}$

Step 2.3.4.2.5

Rewrite the expression.

$\frac{- 1}{2} \cdot \frac{91}{2}$

Step 2.3.4.3

Multiply $\frac{- 1}{2}$ by $\frac{91}{2}$ .

$\frac{- 1 \cdot 91}{2 \cdot 2}$

Step 2.3.4.4

Simplify the expression.

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Step 2.3.4.4.1

Multiply $- 1$ by $91$ .

$\frac{- 91}{2 \cdot 2}$

Step 2.3.4.4.2

Multiply $2$ by $2$ .

$\frac{- 91}{4}$

Step 2.3.4.4.3

Move the negative in front of the fraction.

$- \frac{91}{4}$

Step 2.4

Add the results of the summations.

$\frac{65}{8} - \frac{91}{4}$

Step 2.5

Simplify.

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Step 2.5.1

To write $- \frac{91}{4}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{65}{8} - \frac{91}{4} \cdot \frac{2}{2}$

Step 2.5.2

Write each expression with a common denominator of $8$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.5.2.1

Multiply $\frac{91}{4}$ by $\frac{2}{2}$ .

$\frac{65}{8} - \frac{91 \cdot 2}{4 \cdot 2}$

Step 2.5.2.2

Multiply $4$ by $2$ .

$\frac{65}{8} - \frac{91 \cdot 2}{8}$

Step 2.5.3

Combine the numerators over the common denominator.

$\frac{65 - 91 \cdot 2}{8}$

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Multiply $- 91$ by $2$ .

$\frac{65 - 182}{8}$

Step 2.5.4.2

Subtract $182$ from $65$ .

$\frac{- 117}{8}$

Step 2.5.5

Move the negative in front of the fraction.

$- \frac{117}{8}$

Step 3

Evaluate $\sum_{k = 1}^{4} \frac{5}{8} - \frac{k}{4}$ .

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Step 3.1

Expand the series for each value of $k$ .

$\frac{5}{8} - \frac{1}{4} + \frac{5}{8} - \frac{2}{4} + \frac{5}{8} - \frac{3}{4} + \frac{5}{8} - \frac{4}{4}$

Step 3.2

Simplify.

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Step 3.2.1

Combine the numerators over the common denominator.

$\frac{5 + 5 + 5 + 5}{8} + \frac{- 1 - 2 - 3 - 4}{4}$

Step 3.2.2

Add $5$ and $5$ .

$\frac{10 + 5 + 5}{8} + \frac{- 1 - 2 - 3 - 4}{4}$

Step 3.2.3

Add $10$ and $5$ .

$\frac{15 + 5}{8} + \frac{- 1 - 2 - 3 - 4}{4}$

Step 3.2.4

Add $15$ and $5$ .

$\frac{20}{8} + \frac{- 1 - 2 - 3 - 4}{4}$

Step 3.2.5

Subtract $2$ from $- 1$ .

$\frac{20}{8} + \frac{- 3 - 3 - 4}{4}$

Step 3.2.6

Subtract $3$ from $- 3$ .

$\frac{20}{8} + \frac{- 6 - 4}{4}$

Step 3.2.7

Subtract $4$ from $- 6$ .

$\frac{20}{8} + \frac{- 10}{4}$

Step 3.2.8

Simplify each term.

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Step 3.2.8.1

Cancel the common factor of $20$ and $8$ .

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Step 3.2.8.1.1

Factor $4$ out of $20$ .

$\frac{4 (5)}{8} + \frac{- 10}{4}$

Step 3.2.8.1.2

Cancel the common factors.

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Step 3.2.8.1.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 5}{4 \cdot 2} + \frac{- 10}{4}$

Step 3.2.8.1.2.2

Cancel the common factor.

$\frac{4 \cdot 5}{4 \cdot 2} + \frac{- 10}{4}$

Step 3.2.8.1.2.3

Rewrite the expression.

$\frac{5}{2} + \frac{- 10}{4}$

Step 3.2.8.2

Cancel the common factor of $- 10$ and $4$ .

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Step 3.2.8.2.1

Factor $2$ out of $- 10$ .

$\frac{5}{2} + \frac{2 (- 5)}{4}$

Step 3.2.8.2.2

Cancel the common factors.

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Step 3.2.8.2.2.1

Factor $2$ out of $4$ .

$\frac{5}{2} + \frac{2 \cdot - 5}{2 \cdot 2}$

Step 3.2.8.2.2.2

Cancel the common factor.

$\frac{5}{2} + \frac{2 \cdot - 5}{2 \cdot 2}$

Step 3.2.8.2.2.3

Rewrite the expression.

$\frac{5}{2} + \frac{- 5}{2}$

Step 3.2.8.3

Move the negative in front of the fraction.

$\frac{5}{2} - \frac{5}{2}$

Step 3.2.9

Combine the numerators over the common denominator.

$\frac{5 - 5}{2}$

Step 3.2.10

Subtract $5$ from $5$ .

$\frac{0}{2}$

Step 3.2.11

Divide $0$ by $2$ .

$0$

Step 4

Replace the summations with the values found.

$- \frac{117}{8} - 0$

Step 5

Subtract $0$ from $- \frac{117}{8}$ .

$- \frac{117}{8}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{117}{8}$

Decimal Form:

$- 14.625$

Mixed Number Form:

$- 14 \frac{5}{8}$