Precalculus Examples

$\sum_{k = 15}^{46} \frac{3 - 5 k}{4}$

Step 1

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

$\sum_{k = 15}^{46} \frac{3}{4} - \frac{5 k}{4} = \sum_{k = 1}^{46} \frac{3}{4} - \frac{5 k}{4} - \sum_{k = 1}^{14} \frac{3}{4} - \frac{5 k}{4}$

Step 2

Evaluate $\sum_{k = 1}^{46} \frac{3}{4} - \frac{5 k}{4}$ .

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Step 2.1

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

$\sum_{k = 1}^{46} \frac{3}{4} - \frac{5 k}{4} = \sum_{k = 1}^{46} \frac{3}{4} + \sum_{k = 1}^{46} - \frac{5 k}{4}$

Step 2.2

Evaluate $\sum_{k = 1}^{46} \frac{3}{4}$ .

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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.

$(\frac{3}{4}) (46)$

Step 2.2.3

Simplify.

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Step 2.2.3.1

Cancel the common factor of $2$ .

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Step 2.2.3.1.1

Factor $2$ out of $4$ .

$\frac{3}{2 (2)} \cdot 46$

Step 2.2.3.1.2

Factor $2$ out of $46$ .

$\frac{3}{2 \cdot 2} \cdot (2 \cdot 23)$

Step 2.2.3.1.3

Cancel the common factor.

$\frac{3}{2 \cdot 2} \cdot (2 \cdot 23)$

Step 2.2.3.1.4

Rewrite the expression.

$\frac{3}{2} \cdot 23$

Step 2.2.3.2

Combine $\frac{3}{2}$ and $23$ .

$\frac{3 \cdot 23}{2}$

Step 2.2.3.3

Multiply $3$ by $23$ .

$\frac{69}{2}$

Step 2.3

Evaluate $\sum_{k = 1}^{46} - \frac{5 k}{4}$ .

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Step 2.3.1

Factor $- \frac{5}{4}$ out of the summation.

$(- \frac{5}{4}) \sum_{k = 1}^{46} k$

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.

$(- \frac{5}{4}) (\frac{46 (46 + 1)}{2})$

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 $46$ and $1$ .

$- \frac{5}{4} \cdot \frac{46 \cdot 47}{2}$

Step 2.3.4.1.2

Multiply $46$ by $47$ .

$- \frac{5}{4} \cdot \frac{2162}{2}$

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 $- \frac{5}{4}$ into the numerator.

$\frac{- 5}{4} \cdot \frac{2162}{2}$

Step 2.3.4.2.2

Factor $2$ out of $4$ .

$\frac{- 5}{2 (2)} \cdot \frac{2162}{2}$

Step 2.3.4.2.3

Factor $2$ out of $2162$ .

$\frac{- 5}{2 \cdot 2} \cdot \frac{2 \cdot 1081}{2}$

Step 2.3.4.2.4

Cancel the common factor.

$\frac{- 5}{2 \cdot 2} \cdot \frac{2 \cdot 1081}{2}$

Step 2.3.4.2.5

Rewrite the expression.

$\frac{- 5}{2} \cdot \frac{1081}{2}$

Step 2.3.4.3

Multiply $\frac{- 5}{2}$ by $\frac{1081}{2}$ .

$\frac{- 5 \cdot 1081}{2 \cdot 2}$

Step 2.3.4.4

Simplify the expression.

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Step 2.3.4.4.1

Multiply $- 5$ by $1081$ .

$\frac{- 5405}{2 \cdot 2}$

Step 2.3.4.4.2

Multiply $2$ by $2$ .

$\frac{- 5405}{4}$

Step 2.3.4.4.3

Move the negative in front of the fraction.

$- \frac{5405}{4}$

Step 2.4

Add the results of the summations.

$\frac{69}{2} - \frac{5405}{4}$

Step 2.5

Simplify.

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Step 2.5.1

To write $\frac{69}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{69}{2} \cdot \frac{2}{2} - \frac{5405}{4}$

Step 2.5.2

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

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Step 2.5.2.1

Multiply $\frac{69}{2}$ by $\frac{2}{2}$ .

$\frac{69 \cdot 2}{2 \cdot 2} - \frac{5405}{4}$

Step 2.5.2.2

Multiply $2$ by $2$ .

$\frac{69 \cdot 2}{4} - \frac{5405}{4}$

Step 2.5.3

Combine the numerators over the common denominator.

$\frac{69 \cdot 2 - 5405}{4}$

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Multiply $69$ by $2$ .

$\frac{138 - 5405}{4}$

Step 2.5.4.2

Subtract $5405$ from $138$ .

$\frac{- 5267}{4}$

Step 2.5.5

Move the negative in front of the fraction.

$- \frac{5267}{4}$

Step 3

Evaluate $\sum_{k = 1}^{14} \frac{3}{4} - \frac{5 k}{4}$ .

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Step 3.1

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

$\sum_{k = 1}^{14} \frac{3}{4} - \frac{5 k}{4} = \sum_{k = 1}^{14} \frac{3}{4} + \sum_{k = 1}^{14} - \frac{5 k}{4}$

Step 3.2

Evaluate $\sum_{k = 1}^{14} \frac{3}{4}$ .

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Step 3.2.1

The formula for the summation of a constant is:

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

Step 3.2.2

Substitute the values into the formula.

$(\frac{3}{4}) (14)$

Step 3.2.3

Simplify.

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Step 3.2.3.1

Cancel the common factor of $2$ .

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Step 3.2.3.1.1

Factor $2$ out of $4$ .

$\frac{3}{2 (2)} \cdot 14$

Step 3.2.3.1.2

Factor $2$ out of $14$ .

$\frac{3}{2 \cdot 2} \cdot (2 \cdot 7)$

Step 3.2.3.1.3

Cancel the common factor.

$\frac{3}{2 \cdot 2} \cdot (2 \cdot 7)$

Step 3.2.3.1.4

Rewrite the expression.

$\frac{3}{2} \cdot 7$

Step 3.2.3.2

Combine $\frac{3}{2}$ and $7$ .

$\frac{3 \cdot 7}{2}$

Step 3.2.3.3

Multiply $3$ by $7$ .

$\frac{21}{2}$

Step 3.3

Evaluate $\sum_{k = 1}^{14} - \frac{5 k}{4}$ .

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Step 3.3.1

Factor $- \frac{5}{4}$ out of the summation.

$(- \frac{5}{4}) \sum_{k = 1}^{14} k$

Step 3.3.2

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

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

Step 3.3.3

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

$(- \frac{5}{4}) (\frac{14 (14 + 1)}{2})$

Step 3.3.4

Simplify.

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Step 3.3.4.1

Simplify the expression.

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Step 3.3.4.1.1

Add $14$ and $1$ .

$- \frac{5}{4} \cdot \frac{14 \cdot 15}{2}$

Step 3.3.4.1.2

Multiply $14$ by $15$ .

$- \frac{5}{4} \cdot \frac{210}{2}$

Step 3.3.4.2

Cancel the common factor of $2$ .

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Step 3.3.4.2.1

Move the leading negative in $- \frac{5}{4}$ into the numerator.

$\frac{- 5}{4} \cdot \frac{210}{2}$

Step 3.3.4.2.2

Factor $2$ out of $4$ .

$\frac{- 5}{2 (2)} \cdot \frac{210}{2}$

Step 3.3.4.2.3

Factor $2$ out of $210$ .

$\frac{- 5}{2 \cdot 2} \cdot \frac{2 \cdot 105}{2}$

Step 3.3.4.2.4

Cancel the common factor.

$\frac{- 5}{2 \cdot 2} \cdot \frac{2 \cdot 105}{2}$

Step 3.3.4.2.5

Rewrite the expression.

$\frac{- 5}{2} \cdot \frac{105}{2}$

Step 3.3.4.3

Multiply $\frac{- 5}{2}$ by $\frac{105}{2}$ .

$\frac{- 5 \cdot 105}{2 \cdot 2}$

Step 3.3.4.4

Simplify the expression.

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Step 3.3.4.4.1

Multiply $- 5$ by $105$ .

$\frac{- 525}{2 \cdot 2}$

Step 3.3.4.4.2

Multiply $2$ by $2$ .

$\frac{- 525}{4}$

Step 3.3.4.4.3

Move the negative in front of the fraction.

$- \frac{525}{4}$

Step 3.4

Add the results of the summations.

$\frac{21}{2} - \frac{525}{4}$

Step 3.5

Simplify.

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Step 3.5.1

To write $\frac{21}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{21}{2} \cdot \frac{2}{2} - \frac{525}{4}$

Step 3.5.2

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

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Step 3.5.2.1

Multiply $\frac{21}{2}$ by $\frac{2}{2}$ .

$\frac{21 \cdot 2}{2 \cdot 2} - \frac{525}{4}$

Step 3.5.2.2

Multiply $2$ by $2$ .

$\frac{21 \cdot 2}{4} - \frac{525}{4}$

Step 3.5.3

Combine the numerators over the common denominator.

$\frac{21 \cdot 2 - 525}{4}$

Step 3.5.4

Simplify the numerator.

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Step 3.5.4.1

Multiply $21$ by $2$ .

$\frac{42 - 525}{4}$

Step 3.5.4.2

Subtract $525$ from $42$ .

$\frac{- 483}{4}$

Step 3.5.5

Move the negative in front of the fraction.

$- \frac{483}{4}$

Step 4

Replace the summations with the values found.

$- \frac{5267}{4} + \frac{483}{4}$

Step 5

Simplify.

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Step 5.1

Combine the numerators over the common denominator.

$\frac{- 5267 + 483}{4}$

Step 5.2

Simplify the expression.

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Step 5.2.1

Add $- 5267$ and $483$ .

$\frac{- 4784}{4}$

Step 5.2.2

Divide $- 4784$ by $4$ .

$- 1196$