Calculus Examples

$\sum_{k = 6}^{31} \frac{3 k - 6}{4}$

Step 1

Simplify the summation.

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Step 1.1

Factor $3$ out of $3 k - 6$ .

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Step 1.1.1

Factor $3$ out of $3 k$ .

$\frac{3 (k) - 6}{4}$

Step 1.1.2

Factor $3$ out of $- 6$ .

$\frac{3 k + 3 \cdot - 2}{4}$

Step 1.1.3

Factor $3$ out of $3 k + 3 \cdot - 2$ .

$\frac{3 (k - 2)}{4}$

Step 1.2

Rewrite the summation.

$\sum_{k = 6}^{31} \frac{3 (k - 2)}{4}$

Step 2

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

$\sum_{k = 6}^{31} \frac{3 k}{4} - \frac{3}{2} = \sum_{k = 1}^{31} \frac{3 k}{4} - \frac{3}{2} - \sum_{k = 1}^{5} \frac{3 k}{4} - \frac{3}{2}$

Step 3

Evaluate $\sum_{k = 1}^{31} \frac{3 k}{4} - \frac{3}{2}$ .

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Step 3.1

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

$\sum_{k = 1}^{31} \frac{3 k}{4} - \frac{3}{2} = \sum_{k = 1}^{31} \frac{3 k}{4} + \sum_{k = 1}^{31} - \frac{3}{2}$

Step 3.2

Evaluate $\sum_{k = 1}^{31} \frac{3 k}{4}$ .

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Step 3.2.1

Factor $\frac{3}{4}$ out of the summation.

$(\frac{3}{4}) \sum_{k = 1}^{31} k$

Step 3.2.2

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

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

Step 3.2.3

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

$(\frac{3}{4}) (\frac{31 (31 + 1)}{2})$

Step 3.2.4

Simplify.

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Step 3.2.4.1

Simplify the expression.

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Step 3.2.4.1.1

Add $31$ and $1$ .

$\frac{3}{4} \cdot \frac{31 \cdot 32}{2}$

Step 3.2.4.1.2

Multiply $31$ by $32$ .

$\frac{3}{4} \cdot \frac{992}{2}$

Step 3.2.4.2

Cancel the common factor of $4$ .

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Step 3.2.4.2.1

Factor $4$ out of $992$ .

$\frac{3}{4} \cdot \frac{4 (248)}{2}$

Step 3.2.4.2.2

Cancel the common factor.

$\frac{3}{4} \cdot \frac{4 \cdot 248}{2}$

Step 3.2.4.2.3

Rewrite the expression.

$3 (\frac{248}{2})$

Step 3.2.4.3

Combine $3$ and $\frac{248}{2}$ .

$\frac{3 \cdot 248}{2}$

Step 3.2.4.4

Simplify the expression.

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Step 3.2.4.4.1

Multiply $3$ by $248$ .

$\frac{744}{2}$

Step 3.2.4.4.2

Divide $744$ by $2$ .

$372$

Step 3.3

Evaluate $\sum_{k = 1}^{31} - \frac{3}{2}$ .

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Step 3.3.1

The formula for the summation of a constant is:

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

Step 3.3.2

Substitute the values into the formula.

$(- \frac{3}{2}) (31)$

Step 3.3.3

Simplify.

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Step 3.3.3.1

Multiply $(- \frac{3}{2}) (31)$ .

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Step 3.3.3.1.1

Multiply $31$ by $- 1$ .

$- 31 (\frac{3}{2})$

Step 3.3.3.1.2

Combine $- 31$ and $\frac{3}{2}$ .

$\frac{- 31 \cdot 3}{2}$

Step 3.3.3.1.3

Multiply $- 31$ by $3$ .

$\frac{- 93}{2}$

Step 3.3.3.2

Move the negative in front of the fraction.

$- \frac{93}{2}$

Step 3.4

Add the results of the summations.

$372 - \frac{93}{2}$

Step 3.5

Simplify.

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Step 3.5.1

To write $372$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$372 \cdot \frac{2}{2} - \frac{93}{2}$

Step 3.5.2

Combine $372$ and $\frac{2}{2}$ .

$\frac{372 \cdot 2}{2} - \frac{93}{2}$

Step 3.5.3

Combine the numerators over the common denominator.

$\frac{372 \cdot 2 - 93}{2}$

Step 3.5.4

Simplify the numerator.

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Step 3.5.4.1

Multiply $372$ by $2$ .

$\frac{744 - 93}{2}$

Step 3.5.4.2

Subtract $93$ from $744$ .

$\frac{651}{2}$

Step 4

Evaluate $\sum_{k = 1}^{5} \frac{3 k}{4} - \frac{3}{2}$ .

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Step 4.1

Expand the series for each value of $k$ .

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

Step 4.2

Simplify the expanded form.

$\frac{15}{4}$

Step 5

Replace the summations with the values found.

$\frac{651}{2} - \frac{15}{4}$

Step 6

Simplify.

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Step 6.1

To write $\frac{651}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{651}{2} \cdot \frac{2}{2} - \frac{15}{4}$

Step 6.2

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.2.1

Multiply $\frac{651}{2}$ by $\frac{2}{2}$ .

$\frac{651 \cdot 2}{2 \cdot 2} - \frac{15}{4}$

Step 6.2.2

Multiply $2$ by $2$ .

$\frac{651 \cdot 2}{4} - \frac{15}{4}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{651 \cdot 2 - 15}{4}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $651$ by $2$ .

$\frac{1302 - 15}{4}$

Step 6.4.2

Subtract $15$ from $1302$ .

$\frac{1287}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1287}{4}$

Decimal Form:

$321.75$

Mixed Number Form:

$321 \frac{3}{4}$