Precalculus Examples

$\sum_{k = 1}^{8} 4 {(\frac{2}{3})}^{k - 1}$

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

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Step 2.1

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

$r = \frac{4 {(\frac{2}{3})}^{k + 1 - 1}}{4 {(\frac{2}{3})}^{k - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $4$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{4 {(\frac{2}{3})}^{k + 1 - 1}}{4 {(\frac{2}{3})}^{k - 1}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{(\frac{2}{3})}^{k + 1 - 1}}{{(\frac{2}{3})}^{k - 1}}$

Step 2.2.2

Cancel the common factor of ${(\frac{2}{3})}^{k + 1 - 1}$ and ${(\frac{2}{3})}^{k - 1}$ .

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Step 2.2.2.1

Factor ${(\frac{2}{3})}^{k - 1}$ out of ${(\frac{2}{3})}^{k + 1 - 1}$ .

$r = \frac{{(\frac{2}{3})}^{k - 1} {(\frac{2}{3})}^{k + 0 - (k - 1)}}{{(\frac{2}{3})}^{k - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(\frac{2}{3})}^{k - 1} {(\frac{2}{3})}^{k + 0 - (k - 1)}}{{(\frac{2}{3})}^{k - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{2}{3})}^{k - 1} {(\frac{2}{3})}^{k + 0 - (k - 1)}}{{(\frac{2}{3})}^{k - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(\frac{2}{3})}^{k + 0 - (k - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(\frac{2}{3})}^{k + 0 - (k - 1)}$ by $1$ .

$r = {(\frac{2}{3})}^{k + 0 - (k - 1)}$

Step 2.2.3

Add $k$ and $0$ .

$r = {(\frac{2}{3})}^{k - (k - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {(\frac{2}{3})}^{k - k - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = {(\frac{2}{3})}^{k - k + 1}$

Step 2.2.5

Subtract $k$ from $k$ .

$r = {(\frac{2}{3})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

$r = {(\frac{2}{3})}^{1}$

Step 2.2.7

Simplify.

$r = \frac{2}{3}$

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 $4 {(\frac{2}{3})}^{k - 1}$ .

$a = 4 {(\frac{2}{3})}^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = 4 {(\frac{2}{3})}^{0}$

Step 3.2.2

Apply the product rule to $\frac{2}{3}$ .

$a = 4 \frac{2^{0}}{3^{0}}$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = 4 \frac{1}{3^{0}}$

Step 3.2.4

Anything raised to $0$ is $1$ .

$a = 4 (\frac{1}{1})$

Step 3.2.5

Cancel the common factor of $1$ .

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Step 3.2.5.1

Cancel the common factor.

$a = 4 (\frac{1}{1})$

Step 3.2.5.2

Rewrite the expression.

$a = 4 \cdot 1$

Step 3.2.6

Multiply $4$ by $1$ .

$a = 4$

Step 4

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

$4 \frac{1 - {(\frac{2}{3})}^{8}}{1 - \frac{2}{3}}$

Step 5

Simplify.

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Step 5.1

Multiply the numerator and denominator of the fraction by $3$ .

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Step 5.1.1

Multiply $\frac{1 - {(\frac{2}{3})}^{8}}{1 - \frac{2}{3}}$ by $\frac{3}{3}$ .

$4 (\frac{3}{3} \cdot \frac{1 - {(\frac{2}{3})}^{8}}{1 - \frac{2}{3}})$

Step 5.1.2

Combine.

$4 \frac{3 (1 - {(\frac{2}{3})}^{8})}{3 (1 - \frac{2}{3})}$

Step 5.2

Apply the distributive property.

$4 \frac{3 \cdot 1 + 3 (- {(\frac{2}{3})}^{8})}{3 \cdot 1 + 3 (- \frac{2}{3})}$

Step 5.3

Cancel the common factor of $3$ .

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Step 5.3.1

Move the leading negative in $- \frac{2}{3}$ into the numerator.

$4 \frac{3 \cdot 1 + 3 (- {(\frac{2}{3})}^{8})}{3 \cdot 1 + 3 (\frac{- 2}{3})}$

Step 5.3.2

Cancel the common factor.

$4 \frac{3 \cdot 1 + 3 (- {(\frac{2}{3})}^{8})}{3 \cdot 1 + 3 (\frac{- 2}{3})}$

Step 5.3.3

Rewrite the expression.

$4 \frac{3 \cdot 1 + 3 (- {(\frac{2}{3})}^{8})}{3 \cdot 1 - 2}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $3$ by $1$ .

$4 \frac{3 + 3 (- {(\frac{2}{3})}^{8})}{3 \cdot 1 - 2}$

Step 5.4.2

Apply the product rule to $\frac{2}{3}$ .

$4 \frac{3 + 3 (- \frac{2^{8}}{3^{8}})}{3 \cdot 1 - 2}$

Step 5.4.3

Cancel the common factor of $3$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{2^{8}}{3^{8}}$ into the numerator.

$4 \frac{3 + 3 \frac{- 2^{8}}{3^{8}}}{3 \cdot 1 - 2}$

Step 5.4.3.2

Factor $3$ out of $3^{8}$ .

$4 \frac{3 + 3 \frac{- 2^{8}}{3 \cdot 3^{7}}}{3 \cdot 1 - 2}$

Step 5.4.3.3

Cancel the common factor.

$4 \frac{3 + 3 \frac{- 2^{8}}{3 \cdot 3^{7}}}{3 \cdot 1 - 2}$

Step 5.4.3.4

Rewrite the expression.

$4 \frac{3 + \frac{- 2^{8}}{3^{7}}}{3 \cdot 1 - 2}$

Step 5.4.4

Raise $2$ to the power of $8$ .

$4 \frac{3 + \frac{- 1 \cdot 256}{3^{7}}}{3 \cdot 1 - 2}$

Step 5.4.5

Raise $3$ to the power of $7$ .

$4 \frac{3 + \frac{- 1 \cdot 256}{2187}}{3 \cdot 1 - 2}$

Step 5.4.6

Multiply $- 1$ by $256$ .

$4 \frac{3 + \frac{- 256}{2187}}{3 \cdot 1 - 2}$

Step 5.4.7

Move the negative in front of the fraction.

$4 \frac{3 - \frac{256}{2187}}{3 \cdot 1 - 2}$

Step 5.4.8

To write $3$ as a fraction with a common denominator, multiply by $\frac{2187}{2187}$ .

$4 \frac{3 \cdot \frac{2187}{2187} - \frac{256}{2187}}{3 \cdot 1 - 2}$

Step 5.4.9

Combine $3$ and $\frac{2187}{2187}$ .

$4 \frac{\frac{3 \cdot 2187}{2187} - \frac{256}{2187}}{3 \cdot 1 - 2}$

Step 5.4.10

Combine the numerators over the common denominator.

$4 \frac{\frac{3 \cdot 2187 - 256}{2187}}{3 \cdot 1 - 2}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $3$ by $2187$ .

$4 \frac{\frac{6561 - 256}{2187}}{3 \cdot 1 - 2}$

Step 5.4.11.2

Subtract $256$ from $6561$ .

$4 \frac{\frac{6305}{2187}}{3 \cdot 1 - 2}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $3$ by $1$ .

$4 \frac{\frac{6305}{2187}}{3 - 2}$

Step 5.5.2

Subtract $2$ from $3$ .

$4 \frac{\frac{6305}{2187}}{1}$

Step 5.6

Divide $\frac{6305}{2187}$ by $1$ .

$4 (\frac{6305}{2187})$

Step 5.7

Multiply $4 (\frac{6305}{2187})$ .

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Step 5.7.1

Combine $4$ and $\frac{6305}{2187}$ .

$\frac{4 \cdot 6305}{2187}$

Step 5.7.2

Multiply $4$ by $6305$ .

$\frac{25220}{2187}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{25220}{2187}$

Decimal Form:

$11.53177869 \dots$

Mixed Number Form:

$11 \frac{1163}{2187}$