Calculus Examples

$\sum_{k = 1}^{13} - \frac{9}{2} - \frac{3}{2} k$

Step 1

Simplify the summation.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

Move $3$ to the left of $k$ .

$- \frac{9}{2} - \frac{3 k}{2}$

Step 1.2

Rewrite the summation.

$\sum_{k = 1}^{13} - \frac{9}{2} - \frac{3 k}{2}$

Step 2

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

$\sum_{k = 1}^{13} - \frac{9}{2} - \frac{3 k}{2} = \sum_{k = 1}^{13} - \frac{9}{2} + \sum_{k = 1}^{13} - \frac{3 k}{2}$

Step 3

Evaluate $\sum_{k = 1}^{13} - \frac{9}{2}$ .

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Step 3.1

The formula for the summation of a constant is:

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

Step 3.2

Substitute the values into the formula.

$(- \frac{9}{2}) (13)$

Step 3.3

Simplify.

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Step 3.3.1

Multiply $(- \frac{9}{2}) (13)$ .

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Step 3.3.1.1

Multiply $13$ by $- 1$ .

$- 13 (\frac{9}{2})$

Step 3.3.1.2

Combine $- 13$ and $\frac{9}{2}$ .

$\frac{- 13 \cdot 9}{2}$

Step 3.3.1.3

Multiply $- 13$ by $9$ .

$\frac{- 117}{2}$

Step 3.3.2

Move the negative in front of the fraction.

$- \frac{117}{2}$

Step 4

Evaluate $\sum_{k = 1}^{13} - \frac{3 k}{2}$ .

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Step 4.1

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

$(- \frac{3}{2}) \sum_{k = 1}^{13} k$

Step 4.2

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

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

Step 4.3

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

$(- \frac{3}{2}) (\frac{13 (13 + 1)}{2})$

Step 4.4

Simplify.

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Step 4.4.1

Simplify the expression.

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Step 4.4.1.1

Add $13$ and $1$ .

$- \frac{3}{2} \cdot \frac{13 \cdot 14}{2}$

Step 4.4.1.2

Multiply $13$ by $14$ .

$- \frac{3}{2} \cdot \frac{182}{2}$

Step 4.4.2

Cancel the common factor of $2$ .

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Step 4.4.2.1

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

$\frac{- 3}{2} \cdot \frac{182}{2}$

Step 4.4.2.2

Factor $2$ out of $182$ .

$\frac{- 3}{2} \cdot \frac{2 (91)}{2}$

Step 4.4.2.3

Cancel the common factor.

$\frac{- 3}{2} \cdot \frac{2 \cdot 91}{2}$

Step 4.4.2.4

Rewrite the expression.

$- 3 (\frac{91}{2})$

Step 4.4.3

Combine $- 3$ and $\frac{91}{2}$ .

$\frac{- 3 \cdot 91}{2}$

Step 4.4.4

Simplify the expression.

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Step 4.4.4.1

Multiply $- 3$ by $91$ .

$\frac{- 273}{2}$

Step 4.4.4.2

Move the negative in front of the fraction.

$- \frac{273}{2}$

Step 5

Add the results of the summations.

$- \frac{117}{2} - \frac{273}{2}$

Step 6

Simplify.

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Step 6.1

Combine the numerators over the common denominator.

$\frac{- 117 - 273}{2}$

Step 6.2

Simplify the expression.

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Step 6.2.1

Subtract $273$ from $- 117$ .

$\frac{- 390}{2}$

Step 6.2.2

Divide $- 390$ by $2$ .

$- 195$