Precalculus Examples

$\sum_{i = 0}^{8} 256 + {(\frac{3}{2})}^{i}$

Step 1

Split the summation to make the starting value of $i$ equal to $1$ .

$\sum_{k = 0}^{8} 256 + {(\frac{3}{2})}^{i} = \sum_{k = 1}^{8} 256 + {(\frac{3}{2})}^{i} + \sum_{k = 0}^{0} 256 + {(\frac{3}{2})}^{i}$

Step 2

Evaluate $\sum_{k = 1}^{8} 256 + {(\frac{3}{2})}^{i}$ .

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Step 2.1

Split the summation into smaller summations that fit the summation rules.

$\sum_{k = 1}^{8} 256 + {(\frac{3}{2})}^{i} = \sum_{k = 1}^{8} 256 + \sum_{k = 1}^{8} {(\frac{3}{2})}^{i}$

Step 2.2

Evaluate $\sum_{k = 1}^{8} 256$ .

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Step 2.2.1

The formula for the summation of a constant is:

$\sum_{k = 1}^{n} c = c n$

Step 2.2.2

Substitute the values into the formula.

$(256) (8)$

Step 2.2.3

Multiply $256$ by $8$ .

$2048$

Step 2.3

Evaluate $\sum_{k = 1}^{8} {(\frac{3}{2})}^{i}$ .

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Step 2.3.1

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 2.3.2

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 2.3.2.1

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

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

Step 2.3.2.2

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

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Step 2.3.2.2.1

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

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

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Multiply by $1$ .

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

Step 2.3.2.2.2.2

Cancel the common factor.

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

Step 2.3.2.2.2.3

Rewrite the expression.

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

Step 2.3.2.2.2.4

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

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

Step 2.3.3

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

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Step 2.3.3.1

Substitute $1$ for $i$ into ${(\frac{3}{2})}^{i}$ .

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

Step 2.3.3.2

Simplify.

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

Step 2.3.4

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

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

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

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Step 2.3.5.1.1

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

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

Step 2.3.5.1.2

Combine.

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

Step 2.3.5.2

Apply the distributive property.

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

Step 2.3.5.3

Cancel the common factor of $2$ .

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Step 2.3.5.3.1

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

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

Step 2.3.5.3.2

Cancel the common factor.

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

Step 2.3.5.3.3

Rewrite the expression.

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

Step 2.3.5.4

Simplify the numerator.

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Step 2.3.5.4.1

Multiply $2$ by $1$ .

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

Step 2.3.5.4.2

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

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

Step 2.3.5.4.3

Cancel the common factor of $2$ .

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Step 2.3.5.4.3.1

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

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

Step 2.3.5.4.3.2

Factor $2$ out of $2^{8}$ .

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

Step 2.3.5.4.3.3

Cancel the common factor.

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

Step 2.3.5.4.3.4

Rewrite the expression.

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

Step 2.3.5.4.4

Raise $3$ to the power of $8$ .

$\frac{3}{2} \cdot \frac{2 + \frac{- 1 \cdot 6561}{2^{7}}}{2 \cdot 1 - 3}$

Step 2.3.5.4.5

Raise $2$ to the power of $7$ .

$\frac{3}{2} \cdot \frac{2 + \frac{- 1 \cdot 6561}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.4.6

Multiply $- 1$ by $6561$ .

$\frac{3}{2} \cdot \frac{2 + \frac{- 6561}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.4.7

Move the negative in front of the fraction.

$\frac{3}{2} \cdot \frac{2 - \frac{6561}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.4.8

To write $2$ as a fraction with a common denominator, multiply by $\frac{128}{128}$ .

$\frac{3}{2} \cdot \frac{2 \cdot \frac{128}{128} - \frac{6561}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.4.9

Combine $2$ and $\frac{128}{128}$ .

$\frac{3}{2} \cdot \frac{\frac{2 \cdot 128}{128} - \frac{6561}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.4.10

Combine the numerators over the common denominator.

$\frac{3}{2} \cdot \frac{\frac{2 \cdot 128 - 6561}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.4.11

Simplify the numerator.

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Step 2.3.5.4.11.1

Multiply $2$ by $128$ .

$\frac{3}{2} \cdot \frac{\frac{256 - 6561}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.4.11.2

Subtract $6561$ from $256$ .

$\frac{3}{2} \cdot \frac{\frac{- 6305}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.4.12

Move the negative in front of the fraction.

$\frac{3}{2} \cdot \frac{- \frac{6305}{128}}{2 \cdot 1 - 3}$

Step 2.3.5.5

Simplify the denominator.

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Step 2.3.5.5.1

Multiply $2$ by $1$ .

$\frac{3}{2} \cdot \frac{- \frac{6305}{128}}{2 - 3}$

Step 2.3.5.5.2

Subtract $3$ from $2$ .

$\frac{3}{2} \cdot \frac{- \frac{6305}{128}}{- 1}$

Step 2.3.5.6

Dividing two negative values results in a positive value.

$\frac{3}{2} \cdot \frac{\frac{6305}{128}}{1}$

Step 2.3.5.7

Divide $\frac{6305}{128}$ by $1$ .

$\frac{3}{2} \cdot \frac{6305}{128}$

Step 2.3.5.8

Multiply $\frac{3}{2} \cdot \frac{6305}{128}$ .

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Step 2.3.5.8.1

Multiply $\frac{3}{2}$ by $\frac{6305}{128}$ .

$\frac{3 \cdot 6305}{2 \cdot 128}$

Step 2.3.5.8.2

Multiply $3$ by $6305$ .

$\frac{18915}{2 \cdot 128}$

Step 2.3.5.8.3

Multiply $2$ by $128$ .

$\frac{18915}{256}$

Step 2.4

Add the results of the summations.

$2048 + \frac{18915}{256}$

Step 2.5

Simplify.

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Step 2.5.1

To write $2048$ as a fraction with a common denominator, multiply by $\frac{256}{256}$ .

$2048 \cdot \frac{256}{256} + \frac{18915}{256}$

Step 2.5.2

Combine $2048$ and $\frac{256}{256}$ .

$\frac{2048 \cdot 256}{256} + \frac{18915}{256}$

Step 2.5.3

Combine the numerators over the common denominator.

$\frac{2048 \cdot 256 + 18915}{256}$

Step 2.5.4

Simplify the numerator.

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Step 2.5.4.1

Multiply $2048$ by $256$ .

$\frac{524288 + 18915}{256}$

Step 2.5.4.2

Add $524288$ and $18915$ .

$\frac{543203}{256}$

Step 3

Evaluate $\sum_{k = 0}^{0} 256 + {(\frac{3}{2})}^{i}$ .

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Step 3.1

Expand the series for each value of $i$ .

$256 + {(\frac{3}{2})}^{0}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$256 + \frac{3^{0}}{2^{0}}$

Step 3.2.1.2

Anything raised to $0$ is $1$ .

$256 + \frac{1}{2^{0}}$

Step 3.2.1.3

Anything raised to $0$ is $1$ .

$256 + \frac{1}{1}$

Step 3.2.1.4

Divide $1$ by $1$ .

$256 + 1$

Step 3.2.2

Add $256$ and $1$ .

$257$

Step 4

Replace the summations with the values found.

$\frac{543203}{256} + 257$

Step 5

Simplify.

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Step 5.1

To write $257$ as a fraction with a common denominator, multiply by $\frac{256}{256}$ .

$\frac{543203}{256} + 257 \cdot \frac{256}{256}$

Step 5.2

Combine $257$ and $\frac{256}{256}$ .

$\frac{543203}{256} + \frac{257 \cdot 256}{256}$

Step 5.3

Combine the numerators over the common denominator.

$\frac{543203 + 257 \cdot 256}{256}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $257$ by $256$ .

$\frac{543203 + 65792}{256}$

Step 5.4.2

Add $543203$ and $65792$ .

$\frac{608995}{256}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{608995}{256}$

Decimal Form:

$2378.88671875$

Mixed Number Form:

$2378 \frac{227}{256}$