Precalculus Examples

$\sum_{k = 1}^{8} 7 {(\frac{1}{10})}^{k}$

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 $r = \frac{a_{k + 1}}{a_{k}}$ and simplifying.

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Step 2.1

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

$r = \frac{7 {(\frac{1}{10})}^{k + 1}}{7 {(\frac{1}{10})}^{k}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $7$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{7 {(\frac{1}{10})}^{k + 1}}{7 {(\frac{1}{10})}^{k}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{(\frac{1}{10})}^{k + 1}}{{(\frac{1}{10})}^{k}}$

Step 2.2.2

Cancel the common factor of ${(\frac{1}{10})}^{k + 1}$ and ${(\frac{1}{10})}^{k}$ .

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Step 2.2.2.1

Factor ${(\frac{1}{10})}^{k}$ out of ${(\frac{1}{10})}^{k + 1}$ .

$r = \frac{{(\frac{1}{10})}^{k} \frac{1}{10}}{{(\frac{1}{10})}^{k}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(\frac{1}{10})}^{k} \frac{1}{10}}{{(\frac{1}{10})}^{k} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{1}{10})}^{k} \frac{1}{10}}{{(\frac{1}{10})}^{k} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{\frac{1}{10}}{1}$

Step 2.2.2.2.4

Divide $\frac{1}{10}$ by $1$ .

$r = \frac{1}{10}$

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 $1$ for $k$ into $7 {(\frac{1}{10})}^{k}$ .

$a = 7 {(\frac{1}{10})}^{1}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify.

$a = 7 (\frac{1}{10})$

Step 3.2.2

Combine $7$ and $\frac{1}{10}$ .

$a = \frac{7}{10}$

Step 4

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

$\frac{7}{10} \cdot \frac{1 - {(\frac{1}{10})}^{8}}{1 - \frac{1}{10}}$

Step 5

Simplify.

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Step 5.1

Multiply the numerator and denominator of the fraction by $10$ .

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Step 5.1.1

Multiply $\frac{1 - {(\frac{1}{10})}^{8}}{1 - \frac{1}{10}}$ by $\frac{10}{10}$ .

$\frac{7}{10} (\frac{10}{10} \cdot \frac{1 - {(\frac{1}{10})}^{8}}{1 - \frac{1}{10}})$

Step 5.1.2

Combine.

$\frac{7}{10} \cdot \frac{10 (1 - {(\frac{1}{10})}^{8})}{10 (1 - \frac{1}{10})}$

Step 5.2

Apply the distributive property.

$\frac{7}{10} \cdot \frac{10 \cdot 1 + 10 (- {(\frac{1}{10})}^{8})}{10 \cdot 1 + 10 (- \frac{1}{10})}$

Step 5.3

Cancel the common factor of $10$ .

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Step 5.3.1

Move the leading negative in $- \frac{1}{10}$ into the numerator.

$\frac{7}{10} \cdot \frac{10 \cdot 1 + 10 (- {(\frac{1}{10})}^{8})}{10 \cdot 1 + 10 (\frac{- 1}{10})}$

Step 5.3.2

Cancel the common factor.

$\frac{7}{10} \cdot \frac{10 \cdot 1 + 10 (- {(\frac{1}{10})}^{8})}{10 \cdot 1 + 10 (\frac{- 1}{10})}$

Step 5.3.3

Rewrite the expression.

$\frac{7}{10} \cdot \frac{10 \cdot 1 + 10 (- {(\frac{1}{10})}^{8})}{10 \cdot 1 - 1}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $10$ by $1$ .

$\frac{7}{10} \cdot \frac{10 + 10 (- {(\frac{1}{10})}^{8})}{10 \cdot 1 - 1}$

Step 5.4.2

Apply the product rule to $\frac{1}{10}$ .

$\frac{7}{10} \cdot \frac{10 + 10 (- \frac{1^{8}}{10^{8}})}{10 \cdot 1 - 1}$

Step 5.4.3

Cancel the common factor of $10$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{1^{8}}{10^{8}}$ into the numerator.

$\frac{7}{10} \cdot \frac{10 + 10 \frac{- 1^{8}}{10^{8}}}{10 \cdot 1 - 1}$

Step 5.4.3.2

Factor $10$ out of $10^{8}$ .

$\frac{7}{10} \cdot \frac{10 + 10 \frac{- 1^{8}}{10 \cdot 10^{7}}}{10 \cdot 1 - 1}$

Step 5.4.3.3

Cancel the common factor.

$\frac{7}{10} \cdot \frac{10 + 10 \frac{- 1^{8}}{10 \cdot 10^{7}}}{10 \cdot 1 - 1}$

Step 5.4.3.4

Rewrite the expression.

$\frac{7}{10} \cdot \frac{10 + \frac{- 1^{8}}{10^{7}}}{10 \cdot 1 - 1}$

Step 5.4.4

One to any power is one.

$\frac{7}{10} \cdot \frac{10 + \frac{- 1 \cdot 1}{10^{7}}}{10 \cdot 1 - 1}$

Step 5.4.5

Raise $10$ to the power of $7$ .

$\frac{7}{10} \cdot \frac{10 + \frac{- 1 \cdot 1}{10000000}}{10 \cdot 1 - 1}$

Step 5.4.6

Multiply $- 1$ by $1$ .

$\frac{7}{10} \cdot \frac{10 + \frac{- 1}{10000000}}{10 \cdot 1 - 1}$

Step 5.4.7

Move the negative in front of the fraction.

$\frac{7}{10} \cdot \frac{10 - \frac{1}{10000000}}{10 \cdot 1 - 1}$

Step 5.4.8

To write $10$ as a fraction with a common denominator, multiply by $\frac{10000000}{10000000}$ .

$\frac{7}{10} \cdot \frac{10 \cdot \frac{10000000}{10000000} - \frac{1}{10000000}}{10 \cdot 1 - 1}$

Step 5.4.9

Combine $10$ and $\frac{10000000}{10000000}$ .

$\frac{7}{10} \cdot \frac{\frac{10 \cdot 10000000}{10000000} - \frac{1}{10000000}}{10 \cdot 1 - 1}$

Step 5.4.10

Combine the numerators over the common denominator.

$\frac{7}{10} \cdot \frac{\frac{10 \cdot 10000000 - 1}{10000000}}{10 \cdot 1 - 1}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $10$ by $10000000$ .

$\frac{7}{10} \cdot \frac{\frac{100000000 - 1}{10000000}}{10 \cdot 1 - 1}$

Step 5.4.11.2

Subtract $1$ from $100000000$ .

$\frac{7}{10} \cdot \frac{\frac{99999999}{10000000}}{10 \cdot 1 - 1}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $10$ by $1$ .

$\frac{7}{10} \cdot \frac{\frac{99999999}{10000000}}{10 - 1}$

Step 5.5.2

Subtract $1$ from $10$ .

$\frac{7}{10} \cdot \frac{\frac{99999999}{10000000}}{9}$

Step 5.6

Multiply the numerator by the reciprocal of the denominator.

$\frac{7}{10} (\frac{99999999}{10000000} \cdot \frac{1}{9})$

Step 5.7

Cancel the common factor of $9$ .

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Step 5.7.1

Factor $9$ out of $99999999$ .

$\frac{7}{10} (\frac{9 (11111111)}{10000000} \cdot \frac{1}{9})$

Step 5.7.2

Cancel the common factor.

$\frac{7}{10} (\frac{9 \cdot 11111111}{10000000} \cdot \frac{1}{9})$

Step 5.7.3

Rewrite the expression.

$\frac{7}{10} \cdot \frac{11111111}{10000000}$

Step 5.8

Multiply $\frac{7}{10} \cdot \frac{11111111}{10000000}$ .

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Step 5.8.1

Multiply $\frac{7}{10}$ by $\frac{11111111}{10000000}$ .

$\frac{7 \cdot 11111111}{10 \cdot 10000000}$

Step 5.8.2

Multiply $7$ by $11111111$ .

$\frac{77777777}{10 \cdot 10000000}$

Step 5.8.3

Multiply $10$ by $10000000$ .

$\frac{77777777}{100000000}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{77777777}{100000000}$

Decimal Form:

$0.\overline{7}$