Precalculus Examples

$\sum_{n = 0}^{8} \frac{1}{6^{n}}$

Step 1

The sum of a finite geometric series can be found using the formula $a (\frac{1 - r^{n}}{1 - r})$ where $a$ is the first term and $r$ is the ratio between successive terms.

Step 2

Find the ratio of successive terms by plugging into the formula $r = \frac{a_{n + 1}}{a_{n}}$ and simplifying.

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Step 2.1

Substitute $a_{n}$ and $a_{n + 1}$ into the formula for $r$ .

$r = \frac{\frac{1}{6^{n + 1}}}{\frac{1}{6^{n}}}$

Step 2.2

Simplify.

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Step 2.2.1

Multiply the numerator by the reciprocal of the denominator.

$r = \frac{1}{6^{n + 1}} \cdot 6^{n}$

Step 2.2.2

Combine $\frac{1}{6^{n + 1}}$ and $6^{n}$ .

$r = \frac{6^{n}}{6^{n + 1}}$

Step 2.2.3

Cancel the common factor of $6^{n}$ and $6^{n + 1}$ .

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Step 2.2.3.1

Factor $6^{n + 1}$ out of $6^{n}$ .

$r = \frac{6^{n + 1} \cdot 6^{n - (n + 1)}}{6^{n + 1}}$

Step 2.2.3.2

Cancel the common factors.

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Step 2.2.3.2.1

Multiply by $1$ .

$r = \frac{6^{n + 1} \cdot 6^{n - (n + 1)}}{6^{n + 1} \cdot 1}$

Step 2.2.3.2.2

Cancel the common factor.

$r = \frac{6^{n + 1} \cdot 6^{n - (n + 1)}}{6^{n + 1} \cdot 1}$

Step 2.2.3.2.3

Rewrite the expression.

$r = \frac{6^{n - (n + 1)}}{1}$

Step 2.2.3.2.4

Divide $6^{n - (n + 1)}$ by $1$ .

$r = 6^{n - (n + 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = 6^{n - n - 1 \cdot 1}$

Step 2.2.4.2

Multiply $- 1$ by $1$ .

$r = 6^{n - n - 1}$

Step 2.2.5

Subtract $n$ from $n$ .

$r = 6^{0 - 1}$

Step 2.2.6

Subtract $1$ from $0$ .

$r = 6^{- 1}$

Step 2.2.7

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$r = \frac{1}{6}$

Step 3

Find the first term in the series by substituting in the lower bound and simplifying.

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Step 3.1

Substitute $0$ for $n$ into $\frac{1}{6^{n}}$ .

$a = \frac{1}{6^{0}}$

Step 3.2

Simplify.

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Step 3.2.1

Anything raised to $0$ is $1$ .

$a = \frac{1}{1}$

Step 3.2.2

Divide $1$ by $1$ .

$a = 1$

Step 4

Substitute the values of the ratio, first term, and number of terms into the sum formula.

$1 \frac{1 - {(\frac{1}{6})}^{9}}{1 - \frac{1}{6}}$

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1 - {(\frac{1}{6})}^{9}}{1 - \frac{1}{6}}$ by $1$ .

$\frac{1 - {(\frac{1}{6})}^{9}}{1 - \frac{1}{6}}$

Step 5.2

Multiply the numerator and denominator of the fraction by $6$ .

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Step 5.2.1

Multiply $\frac{1 - {(\frac{1}{6})}^{9}}{1 - \frac{1}{6}}$ by $\frac{6}{6}$ .

$\frac{6}{6} \cdot \frac{1 - {(\frac{1}{6})}^{9}}{1 - \frac{1}{6}}$

Step 5.2.2

Combine.

$\frac{6 (1 - {(\frac{1}{6})}^{9})}{6 (1 - \frac{1}{6})}$

Step 5.3

Apply the distributive property.

$\frac{6 \cdot 1 + 6 (- {(\frac{1}{6})}^{9})}{6 \cdot 1 + 6 (- \frac{1}{6})}$

Step 5.4

Cancel the common factor of $6$ .

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Step 5.4.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$\frac{6 \cdot 1 + 6 (- {(\frac{1}{6})}^{9})}{6 \cdot 1 + 6 (\frac{- 1}{6})}$

Step 5.4.2

Cancel the common factor.

$\frac{6 \cdot 1 + 6 (- {(\frac{1}{6})}^{9})}{6 \cdot 1 + 6 (\frac{- 1}{6})}$

Step 5.4.3

Rewrite the expression.

$\frac{6 \cdot 1 + 6 (- {(\frac{1}{6})}^{9})}{6 \cdot 1 - 1}$

Step 5.5

Simplify the numerator.

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Step 5.5.1

Multiply $6$ by $1$ .

$\frac{6 + 6 (- {(\frac{1}{6})}^{9})}{6 \cdot 1 - 1}$

Step 5.5.2

Apply the product rule to $\frac{1}{6}$ .

$\frac{6 + 6 (- \frac{1^{9}}{6^{9}})}{6 \cdot 1 - 1}$

Step 5.5.3

Cancel the common factor of $6$ .

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Step 5.5.3.1

Move the leading negative in $- \frac{1^{9}}{6^{9}}$ into the numerator.

$\frac{6 + 6 \frac{- 1^{9}}{6^{9}}}{6 \cdot 1 - 1}$

Step 5.5.3.2

Factor $6$ out of $6^{9}$ .

$\frac{6 + 6 \frac{- 1^{9}}{6 \cdot 6^{8}}}{6 \cdot 1 - 1}$

Step 5.5.3.3

Cancel the common factor.

$\frac{6 + 6 \frac{- 1^{9}}{6 \cdot 6^{8}}}{6 \cdot 1 - 1}$

Step 5.5.3.4

Rewrite the expression.

$\frac{6 + \frac{- 1^{9}}{6^{8}}}{6 \cdot 1 - 1}$

Step 5.5.4

One to any power is one.

$\frac{6 + \frac{- 1 \cdot 1}{6^{8}}}{6 \cdot 1 - 1}$

Step 5.5.5

Raise $6$ to the power of $8$ .

$\frac{6 + \frac{- 1 \cdot 1}{1679616}}{6 \cdot 1 - 1}$

Step 5.5.6

Multiply $- 1$ by $1$ .

$\frac{6 + \frac{- 1}{1679616}}{6 \cdot 1 - 1}$

Step 5.5.7

Move the negative in front of the fraction.

$\frac{6 - \frac{1}{1679616}}{6 \cdot 1 - 1}$

Step 5.5.8

To write $6$ as a fraction with a common denominator, multiply by $\frac{1679616}{1679616}$ .

$\frac{6 \cdot \frac{1679616}{1679616} - \frac{1}{1679616}}{6 \cdot 1 - 1}$

Step 5.5.9

Combine $6$ and $\frac{1679616}{1679616}$ .

$\frac{\frac{6 \cdot 1679616}{1679616} - \frac{1}{1679616}}{6 \cdot 1 - 1}$

Step 5.5.10

Combine the numerators over the common denominator.

$\frac{\frac{6 \cdot 1679616 - 1}{1679616}}{6 \cdot 1 - 1}$

Step 5.5.11

Simplify the numerator.

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Step 5.5.11.1

Multiply $6$ by $1679616$ .

$\frac{\frac{10077696 - 1}{1679616}}{6 \cdot 1 - 1}$

Step 5.5.11.2

Subtract $1$ from $10077696$ .

$\frac{\frac{10077695}{1679616}}{6 \cdot 1 - 1}$

Step 5.6

Simplify the denominator.

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Step 5.6.1

Multiply $6$ by $1$ .

$\frac{\frac{10077695}{1679616}}{6 - 1}$

Step 5.6.2

Subtract $1$ from $6$ .

$\frac{\frac{10077695}{1679616}}{5}$

Step 5.7

Multiply the numerator by the reciprocal of the denominator.

$\frac{10077695}{1679616} \cdot \frac{1}{5}$

Step 5.8

Cancel the common factor of $5$ .

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Step 5.8.1

Factor $5$ out of $10077695$ .

$\frac{5 (2015539)}{1679616} \cdot \frac{1}{5}$

Step 5.8.2

Cancel the common factor.

$\frac{5 \cdot 2015539}{1679616} \cdot \frac{1}{5}$

Step 5.8.3

Rewrite the expression.

$\frac{2015539}{1679616}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{2015539}{1679616}$

Decimal Form:

$1.19999988 \dots$

Mixed Number Form:

$1 \frac{335923}{1679616}$