Precalculus Examples

$\sum_{n = 0}^{15} 2 (\frac{4}{3}) n$

Step 1

Simplify the summation.

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Step 1.1

Multiply $2 (\frac{4}{3})$ .

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Step 1.1.1

Combine $2$ and $\frac{4}{3}$ .

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

Step 1.1.2

Multiply $2$ by $4$ .

$\frac{8}{3} \cdot n$

Step 1.2

Combine $\frac{8}{3}$ and $n$ .

$\frac{8 n}{3}$

Step 1.3

Rewrite the summation.

$\sum_{n = 0}^{15} \frac{8 n}{3}$

Step 2

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

$\sum_{n = 0}^{15} \frac{8 n}{3} = \sum_{n = 1}^{15} \frac{8 n}{3} + \sum_{n = 0}^{0} \frac{8 n}{3}$

Step 3

Evaluate $\sum_{n = 1}^{15} \frac{8 n}{3}$ .

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Step 3.1

Factor $\frac{8}{3}$ out of the summation.

$(\frac{8}{3}) \sum_{n = 1}^{15} n$

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.

$(\frac{8}{3}) (\frac{15 (15 + 1)}{2})$

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 $15$ and $1$ .

$\frac{8}{3} \cdot \frac{15 \cdot 16}{2}$

Step 3.4.1.2

Multiply $15$ by $16$ .

$\frac{8}{3} \cdot \frac{240}{2}$

Step 3.4.2

Cancel the common factor of $2$ .

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Step 3.4.2.1

Factor $2$ out of $8$ .

$\frac{2 (4)}{3} \cdot \frac{240}{2}$

Step 3.4.2.2

Cancel the common factor.

$\frac{2 \cdot 4}{3} \cdot \frac{240}{2}$

Step 3.4.2.3

Rewrite the expression.

$\frac{4}{3} \cdot 240$

Step 3.4.3

Cancel the common factor of $3$ .

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Step 3.4.3.1

Factor $3$ out of $240$ .

$\frac{4}{3} \cdot (3 (80))$

Step 3.4.3.2

Cancel the common factor.

$\frac{4}{3} \cdot (3 \cdot 80)$

Step 3.4.3.3

Rewrite the expression.

$4 \cdot 80$

Step 3.4.4

Multiply $4$ by $80$ .

$320$

Step 4

Evaluate $\sum_{n = 0}^{0} \frac{8 n}{3}$ .

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Step 4.1

Expand the series for each value of $n$ .

$\frac{8 \cdot 0}{3}$

Step 4.2

Simplify.

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Step 4.2.1

Cancel the common factor of $0$ and $3$ .

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Step 4.2.1.1

Factor $3$ out of $8 \cdot 0$ .

$\frac{3 (8 \cdot 0)}{3}$

Step 4.2.1.2

Cancel the common factors.

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Step 4.2.1.2.1

Factor $3$ out of $3$ .

$\frac{3 (8 \cdot 0)}{3 (1)}$

Step 4.2.1.2.2

Cancel the common factor.

$\frac{3 (8 \cdot 0)}{3 \cdot 1}$

Step 4.2.1.2.3

Rewrite the expression.

$\frac{8 \cdot 0}{1}$

Step 4.2.1.2.4

Divide $8 \cdot 0$ by $1$ .

$8 \cdot 0$

Step 4.2.2

Multiply $8$ by $0$ .

$0$

Step 5

Replace the summations with the values found.

$320 + 0$

Step 6

Add $320$ and $0$ .

$320$