Calculus Examples

$\sum_{n = 18}^{64} \frac{1}{2} n$

Step 1

Simplify the summation.

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Step 1.1

Combine $\frac{1}{2}$ and $n$ .

$\frac{n}{2}$

Step 1.2

Rewrite the summation.

$\sum_{n = 18}^{64} \frac{n}{2}$

Step 2

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

$\sum_{n = 18}^{64} \frac{n}{2} = \sum_{n = 1}^{64} \frac{n}{2} - \sum_{n = 1}^{17} \frac{n}{2}$

Step 3

Evaluate $\sum_{n = 1}^{64} \frac{n}{2}$ .

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Step 3.1

Factor $\frac{1}{2}$ out of the summation.

$(\frac{1}{2}) \sum_{n = 1}^{64} n$

Step 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

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

$(\frac{1}{2}) (\frac{64 (64 + 1)}{2})$

Step 3.4

Simplify.

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Step 3.4.1

Simplify the expression.

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Step 3.4.1.1

Add $64$ and $1$ .

$\frac{1}{2} \cdot \frac{64 \cdot 65}{2}$

Step 3.4.1.2

Multiply $64$ by $65$ .

$\frac{1}{2} \cdot \frac{4160}{2}$

Step 3.4.2

Cancel the common factor of $2$ .

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Step 3.4.2.1

Factor $2$ out of $4160$ .

$\frac{1}{2} \cdot \frac{2 (2080)}{2}$

Step 3.4.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot 2080}{2}$

Step 3.4.2.3

Rewrite the expression.

$\frac{2080}{2}$

Step 3.4.3

Divide $2080$ by $2$ .

$1040$

Step 4

Evaluate $\sum_{n = 1}^{17} \frac{n}{2}$ .

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Step 4.1

Factor $\frac{1}{2}$ out of the summation.

$(\frac{1}{2}) \sum_{n = 1}^{17} n$

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.

$(\frac{1}{2}) (\frac{17 (17 + 1)}{2})$

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 $17$ and $1$ .

$\frac{1}{2} \cdot \frac{17 \cdot 18}{2}$

Step 4.4.1.2

Multiply $17$ by $18$ .

$\frac{1}{2} \cdot \frac{306}{2}$

Step 4.4.2

Cancel the common factor of $2$ .

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Step 4.4.2.1

Factor $2$ out of $306$ .

$\frac{1}{2} \cdot \frac{2 (153)}{2}$

Step 4.4.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot 153}{2}$

Step 4.4.2.3

Rewrite the expression.

$\frac{153}{2}$

Step 5

Replace the summations with the values found.

$1040 - \frac{153}{2}$

Step 6

Simplify.

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Step 6.1

To write $1040$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$1040 \cdot \frac{2}{2} - \frac{153}{2}$

Step 6.2

Combine $1040$ and $\frac{2}{2}$ .

$\frac{1040 \cdot 2}{2} - \frac{153}{2}$

Step 6.3

Combine the numerators over the common denominator.

$\frac{1040 \cdot 2 - 153}{2}$

Step 6.4

Simplify the numerator.

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Step 6.4.1

Multiply $1040$ by $2$ .

$\frac{2080 - 153}{2}$

Step 6.4.2

Subtract $153$ from $2080$ .

$\frac{1927}{2}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$\frac{1927}{2}$

Decimal Form:

$963.5$

Mixed Number Form:

$963 \frac{1}{2}$