Precalculus Examples

$\sum_{n = 0}^{23} 2 + \frac{1}{2} n$

Step 1

Simplify the summation.

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Step 1.1

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

$2 + \frac{n}{2}$

Step 1.2

Rewrite the summation.

$\sum_{n = 0}^{23} 2 + \frac{n}{2}$

Step 2

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

$\sum_{n = 0}^{23} 2 + \frac{n}{2} = \sum_{n = 1}^{23} 2 + \frac{n}{2} + \sum_{n = 0}^{0} 2 + \frac{n}{2}$

Step 3

Evaluate $\sum_{n = 1}^{23} 2 + \frac{n}{2}$ .

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Step 3.1

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

$\sum_{n = 1}^{23} 2 + \frac{n}{2} = \sum_{n = 1}^{23} 2 + \sum_{n = 1}^{23} \frac{n}{2}$

Step 3.2

Evaluate $\sum_{n = 1}^{23} 2$ .

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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.

$(2) (23)$

Step 3.2.3

Multiply $2$ by $23$ .

$46$

Step 3.3

Evaluate $\sum_{n = 1}^{23} \frac{n}{2}$ .

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Step 3.3.1

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

$(\frac{1}{2}) \sum_{n = 1}^{23} n$

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.

$(\frac{1}{2}) (\frac{23 (23 + 1)}{2})$

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 $23$ and $1$ .

$\frac{1}{2} \cdot \frac{23 \cdot 24}{2}$

Step 3.3.4.1.2

Multiply $23$ by $24$ .

$\frac{1}{2} \cdot \frac{552}{2}$

Step 3.3.4.2

Cancel the common factor of $2$ .

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Step 3.3.4.2.1

Factor $2$ out of $552$ .

$\frac{1}{2} \cdot \frac{2 (276)}{2}$

Step 3.3.4.2.2

Cancel the common factor.

$\frac{1}{2} \cdot \frac{2 \cdot 276}{2}$

Step 3.3.4.2.3

Rewrite the expression.

$\frac{276}{2}$

Step 3.3.4.3

Divide $276$ by $2$ .

$138$

Step 3.4

Add the results of the summations.

$46 + 138$

Step 3.5

Add $46$ and $138$ .

$184$

Step 4

Evaluate $\sum_{n = 0}^{0} 2 + \frac{n}{2}$ .

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Step 4.1

Expand the series for each value of $n$ .

$2 + \frac{0}{2}$

Step 4.2

Simplify.

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Step 4.2.1

Divide $0$ by $2$ .

$2 + 0$

Step 4.2.2

Add $2$ and $0$ .

$2$

Step 5

Replace the summations with the values found.

$184 + 2$

Step 6

Add $184$ and $2$ .

$186$