Precalculus Examples

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

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

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Step 2.1

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

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

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.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

Cancel the common factor of ${(\frac{7}{10})}^{i + 1 - 1}$ and ${(\frac{7}{10})}^{i - 1}$ .

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Step 2.2.2.1

Factor ${(\frac{7}{10})}^{i - 1}$ out of ${(\frac{7}{10})}^{i + 1 - 1}$ .

$r = \frac{{(\frac{7}{10})}^{i - 1} {(\frac{7}{10})}^{i + 0 - (i - 1)}}{{(\frac{7}{10})}^{i - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(\frac{7}{10})}^{i - 1} {(\frac{7}{10})}^{i + 0 - (i - 1)}}{{(\frac{7}{10})}^{i - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{7}{10})}^{i - 1} {(\frac{7}{10})}^{i + 0 - (i - 1)}}{{(\frac{7}{10})}^{i - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(\frac{7}{10})}^{i + 0 - (i - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(\frac{7}{10})}^{i + 0 - (i - 1)}$ by $1$ .

$r = {(\frac{7}{10})}^{i + 0 - (i - 1)}$

Step 2.2.3

Add $i$ and $0$ .

$r = {(\frac{7}{10})}^{i - (i - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {(\frac{7}{10})}^{i - i - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.5

Subtract $i$ from $i$ .

$r = {(\frac{7}{10})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

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

Step 2.2.7

Simplify.

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

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 $i$ into $7 {(\frac{7}{10})}^{i - 1}$ .

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

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = 7 {(\frac{7}{10})}^{0}$

Step 3.2.2

Apply the product rule to $\frac{7}{10}$ .

$a = 7 \frac{7^{0}}{10^{0}}$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = 7 \frac{1}{10^{0}}$

Step 3.2.4

Anything raised to $0$ is $1$ .

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

Step 3.2.5

Cancel the common factor of $1$ .

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Step 3.2.5.1

Cancel the common factor.

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

Step 3.2.5.2

Rewrite the expression.

$a = 7 \cdot 1$

Step 3.2.6

Multiply $7$ by $1$ .

$a = 7$

Step 4

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

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

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

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

Step 5.1.2

Combine.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Cancel the common factor of $10$ .

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Step 5.3.1

Move the leading negative in $- \frac{7}{10}$ into the numerator.

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

Step 5.3.2

Cancel the common factor.

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

Step 5.3.3

Rewrite the expression.

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

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $10$ by $1$ .

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

Step 5.4.2

Apply the product rule to $\frac{7}{10}$ .

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

Step 5.4.3

Cancel the common factor of $10$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{7^{8}}{10^{8}}$ into the numerator.

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

Step 5.4.3.2

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

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

Step 5.4.3.3

Cancel the common factor.

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

Step 5.4.3.4

Rewrite the expression.

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

Step 5.4.4

Raise $7$ to the power of $8$ .

$7 \frac{10 + \frac{- 1 \cdot 5764801}{10^{7}}}{10 \cdot 1 - 7}$

Step 5.4.5

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

$7 \frac{10 + \frac{- 1 \cdot 5764801}{10000000}}{10 \cdot 1 - 7}$

Step 5.4.6

Multiply $- 1$ by $5764801$ .

$7 \frac{10 + \frac{- 5764801}{10000000}}{10 \cdot 1 - 7}$

Step 5.4.7

Move the negative in front of the fraction.

$7 \frac{10 - \frac{5764801}{10000000}}{10 \cdot 1 - 7}$

Step 5.4.8

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

$7 \frac{10 \cdot \frac{10000000}{10000000} - \frac{5764801}{10000000}}{10 \cdot 1 - 7}$

Step 5.4.9

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

$7 \frac{\frac{10 \cdot 10000000}{10000000} - \frac{5764801}{10000000}}{10 \cdot 1 - 7}$

Step 5.4.10

Combine the numerators over the common denominator.

$7 \frac{\frac{10 \cdot 10000000 - 5764801}{10000000}}{10 \cdot 1 - 7}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $10$ by $10000000$ .

$7 \frac{\frac{100000000 - 5764801}{10000000}}{10 \cdot 1 - 7}$

Step 5.4.11.2

Subtract $5764801$ from $100000000$ .

$7 \frac{\frac{94235199}{10000000}}{10 \cdot 1 - 7}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $10$ by $1$ .

$7 \frac{\frac{94235199}{10000000}}{10 - 7}$

Step 5.5.2

Subtract $7$ from $10$ .

$7 \frac{\frac{94235199}{10000000}}{3}$

Step 5.6

Multiply the numerator by the reciprocal of the denominator.

$7 (\frac{94235199}{10000000} \cdot \frac{1}{3})$

Step 5.7

Cancel the common factor of $3$ .

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Step 5.7.1

Factor $3$ out of $94235199$ .

$7 (\frac{3 (31411733)}{10000000} \cdot \frac{1}{3})$

Step 5.7.2

Cancel the common factor.

$7 (\frac{3 \cdot 31411733}{10000000} \cdot \frac{1}{3})$

Step 5.7.3

Rewrite the expression.

$7 (\frac{31411733}{10000000})$

Step 5.8

Multiply $7 (\frac{31411733}{10000000})$ .

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Step 5.8.1

Combine $7$ and $\frac{31411733}{10000000}$ .

$\frac{7 \cdot 31411733}{10000000}$

Step 5.8.2

Multiply $7$ by $31411733$ .

$\frac{219882131}{10000000}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{219882131}{10000000}$

Decimal Form:

$21.9882131$

Mixed Number Form:

$21 \frac{9882131}{10000000}$