Precalculus Examples

$\sum_{n = 0}^{20} 3 {(\frac{3}{2})}^{n}$

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 $r$ .

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

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{\frac{3}{2}}{1}$

Step 2.2.2.2.4

Divide $\frac{3}{2}$ by $1$ .

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

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 $3 {(\frac{3}{2})}^{n}$ .

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

Step 3.2

Simplify.

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Step 3.2.1

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

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

Step 3.2.2

Anything raised to $0$ is $1$ .

$a = 3 \frac{1}{2^{0}}$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = 3 (\frac{1}{1})$

Step 3.2.4

Cancel the common factor of $1$ .

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Step 3.2.4.1

Cancel the common factor.

$a = 3 (\frac{1}{1})$

Step 3.2.4.2

Rewrite the expression.

$a = 3 \cdot 1$

Step 3.2.5

Multiply $3$ by $1$ .

$a = 3$

Step 4

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

$3 \frac{1 - {(\frac{3}{2})}^{21}}{1 - \frac{3}{2}}$

Step 5

Simplify.

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Step 5.1

Multiply the numerator and denominator of the fraction by $2$ .

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Step 5.1.1

Multiply $\frac{1 - {(\frac{3}{2})}^{21}}{1 - \frac{3}{2}}$ by $\frac{2}{2}$ .

$3 (\frac{2}{2} \cdot \frac{1 - {(\frac{3}{2})}^{21}}{1 - \frac{3}{2}})$

Step 5.1.2

Combine.

$3 \frac{2 (1 - {(\frac{3}{2})}^{21})}{2 (1 - \frac{3}{2})}$

Step 5.2

Apply the distributive property.

$3 \frac{2 \cdot 1 + 2 (- {(\frac{3}{2})}^{21})}{2 \cdot 1 + 2 (- \frac{3}{2})}$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

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

$3 \frac{2 \cdot 1 + 2 (- {(\frac{3}{2})}^{21})}{2 \cdot 1 + 2 (\frac{- 3}{2})}$

Step 5.3.2

Cancel the common factor.

$3 \frac{2 \cdot 1 + 2 (- {(\frac{3}{2})}^{21})}{2 \cdot 1 + 2 (\frac{- 3}{2})}$

Step 5.3.3

Rewrite the expression.

$3 \frac{2 \cdot 1 + 2 (- {(\frac{3}{2})}^{21})}{2 \cdot 1 - 3}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $2$ by $1$ .

$3 \frac{2 + 2 (- {(\frac{3}{2})}^{21})}{2 \cdot 1 - 3}$

Step 5.4.2

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

$3 \frac{2 + 2 (- \frac{3^{21}}{2^{21}})}{2 \cdot 1 - 3}$

Step 5.4.3

Cancel the common factor of $2$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{3^{21}}{2^{21}}$ into the numerator.

$3 \frac{2 + 2 \frac{- 3^{21}}{2^{21}}}{2 \cdot 1 - 3}$

Step 5.4.3.2

Factor $2$ out of $2^{21}$ .

$3 \frac{2 + 2 \frac{- 3^{21}}{2 \cdot 2^{20}}}{2 \cdot 1 - 3}$

Step 5.4.3.3

Cancel the common factor.

$3 \frac{2 + 2 \frac{- 3^{21}}{2 \cdot 2^{20}}}{2 \cdot 1 - 3}$

Step 5.4.3.4

Rewrite the expression.

$3 \frac{2 + \frac{- 3^{21}}{2^{20}}}{2 \cdot 1 - 3}$

Step 5.4.4

Raise $3$ to the power of $21$ .

$3 \frac{2 + \frac{- 1 \cdot 10460353203}{2^{20}}}{2 \cdot 1 - 3}$

Step 5.4.5

Raise $2$ to the power of $20$ .

$3 \frac{2 + \frac{- 1 \cdot 10460353203}{1048576}}{2 \cdot 1 - 3}$

Step 5.4.6

Multiply $- 1$ by $10460353203$ .

$3 \frac{2 + \frac{- 10460353203}{1048576}}{2 \cdot 1 - 3}$

Step 5.4.7

Move the negative in front of the fraction.

$3 \frac{2 - \frac{10460353203}{1048576}}{2 \cdot 1 - 3}$

Step 5.4.8

To write $2$ as a fraction with a common denominator, multiply by $\frac{1048576}{1048576}$ .

$3 \frac{2 \cdot \frac{1048576}{1048576} - \frac{10460353203}{1048576}}{2 \cdot 1 - 3}$

Step 5.4.9

Combine $2$ and $\frac{1048576}{1048576}$ .

$3 \frac{\frac{2 \cdot 1048576}{1048576} - \frac{10460353203}{1048576}}{2 \cdot 1 - 3}$

Step 5.4.10

Combine the numerators over the common denominator.

$3 \frac{\frac{2 \cdot 1048576 - 10460353203}{1048576}}{2 \cdot 1 - 3}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $2$ by $1048576$ .

$3 \frac{\frac{2097152 - 10460353203}{1048576}}{2 \cdot 1 - 3}$

Step 5.4.11.2

Subtract $10460353203$ from $2097152$ .

$3 \frac{\frac{- 10458256051}{1048576}}{2 \cdot 1 - 3}$

Step 5.4.12

Move the negative in front of the fraction.

$3 \frac{- \frac{10458256051}{1048576}}{2 \cdot 1 - 3}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $2$ by $1$ .

$3 \frac{- \frac{10458256051}{1048576}}{2 - 3}$

Step 5.5.2

Subtract $3$ from $2$ .

$3 \frac{- \frac{10458256051}{1048576}}{- 1}$

Step 5.6

Dividing two negative values results in a positive value.

$3 \frac{\frac{10458256051}{1048576}}{1}$

Step 5.7

Divide $\frac{10458256051}{1048576}$ by $1$ .

$3 (\frac{10458256051}{1048576})$

Step 5.8

Multiply $3 (\frac{10458256051}{1048576})$ .

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Step 5.8.1

Combine $3$ and $\frac{10458256051}{1048576}$ .

$\frac{3 \cdot 10458256051}{1048576}$

Step 5.8.2

Multiply $3$ by $10458256051$ .

$\frac{31374768153}{1048576}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{31374768153}{1048576}$

Decimal Form:

$29921.31057071 \dots$

Mixed Number Form:

$29921 \frac{325657}{1048576}$