Precalculus Examples

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

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

Step 1

Split the summation to make the starting value of

i

$i$ equal to

1

$1$ .

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

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

Step 2

Evaluate

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

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

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Step 2.1

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

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

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

Step 2.2

Evaluate

8 \sum k = 1 256

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

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Step 2.2.1

The formula for the summation of a constant is:

n \sum k = 1 c = c n

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

Step 2.2.2

Substitute the values into the formula.

(256) (8)

$(256) (8)$

Step 2.2.3

Multiply

256

$256$ by

8

$8$ .

2048

$2048$

2048

$2048$

Step 2.3

Evaluate

8 \sum k = 1 {(\frac{3}{2})}^{i}

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

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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})

$a (\frac{1 - r^{n}}{1 - r})$ where

a

$a$ is the first term and

r

$r$ 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}}

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

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Step 2.3.2.1

Substitute

a_{k}

$a_{k}$ and

a_{k + 1}

$a_{k + 1}$ into the formula for

r

$r$ .

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

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

Step 2.3.2.2

Cancel the common factor of

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

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

{(\frac{3}{2})}^{i}

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

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Step 2.3.2.2.1

Factor

{(\frac{3}{2})}^{i}

${(\frac{3}{2})}^{i}$ out of

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

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

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

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

Step 2.3.2.2.2

Cancel the common factors.

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Step 2.3.2.2.2.1

Multiply by

1

$1$ .

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

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

Step 2.3.2.2.2.3

Rewrite the expression.

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

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

Step 2.3.2.2.2.4

Divide

\frac{3}{2}

$\frac{3}{2}$ by

1

$1$ .

r = \frac{3}{2}

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

r = \frac{3}{2}

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

r = \frac{3}{2}

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

r = \frac{3}{2}

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

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

$1$ for

i

$i$ into

{(\frac{3}{2})}^{i}

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

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

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

Step 2.3.3.2

Simplify.

a = \frac{3}{2}

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

a = \frac{3}{2}

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

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}}

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

Step 2.3.5

Simplify.

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Step 2.3.5.1

Multiply the numerator and denominator of the fraction by

2

$2$ .

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Step 2.3.5.1.1

Multiply

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

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

\frac{2}{2}

$\frac{2}{2}$ .

\frac{3}{2} ⎛ ⎜ ⎜ ⎝ \frac{2}{2} \cdot \frac{1 - {(\frac{3}{2})}^{8}}{1 - \frac{3}{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})}

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

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

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

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})}

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

Step 2.3.5.3

Cancel the common factor of

2

$2$ .

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Step 2.3.5.3.1

Move the leading negative in

- \frac{3}{2}

$- \frac{3}{2}$ into the numerator.

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

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

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:

$2378 \frac{227}{256}$