Calculus Examples

$\sum_{i = 1}^{9} {(- \frac{1}{4})}^{i}$

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 $r$ .

$r = \frac{{(- \frac{1}{4})}^{i + 1}}{{(- \frac{1}{4})}^{i}}$

Step 2.2

Cancel the common factor of ${(- \frac{1}{4})}^{i + 1}$ and ${(- \frac{1}{4})}^{i}$ .

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Step 2.2.1

Factor ${(- \frac{1}{4})}^{i}$ out of ${(- \frac{1}{4})}^{i + 1}$ .

$r = \frac{{(- \frac{1}{4})}^{i} (- \frac{1}{4})}{{(- \frac{1}{4})}^{i}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

$r = \frac{{(- \frac{1}{4})}^{i} (- \frac{1}{4})}{{(- \frac{1}{4})}^{i} \cdot 1}$

Step 2.2.2.2

Cancel the common factor.

$r = \frac{{(- \frac{1}{4})}^{i} (- \frac{1}{4})}{{(- \frac{1}{4})}^{i} \cdot 1}$

Step 2.2.2.3

Rewrite the expression.

$r = \frac{- \frac{1}{4}}{1}$

Step 2.2.2.4

Divide $- \frac{1}{4}$ by $1$ .

$r = - \frac{1}{4}$

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 ${(- \frac{1}{4})}^{i}$ .

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

Step 3.2

Simplify.

$a = - \frac{1}{4}$

Step 4

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

$- \frac{1}{4} \cdot \frac{1 - {(- \frac{1}{4})}^{9}}{1 - - \frac{1}{4}}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.1.1.1

Apply the product rule to $- \frac{1}{4}$ .

$- \frac{1}{4} \cdot \frac{1 - ({(- 1)}^{9} {(\frac{1}{4})}^{9})}{1 - - \frac{1}{4}}$

Step 5.1.1.2

Apply the product rule to $\frac{1}{4}$ .

$- \frac{1}{4} \cdot \frac{1 - ({(- 1)}^{9} \frac{1^{9}}{4^{9}})}{1 - - \frac{1}{4}}$

Step 5.1.2

Multiply $- 1$ by ${(- 1)}^{9}$ by adding the exponents.

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Step 5.1.2.1

Move ${(- 1)}^{9}$ .

$- \frac{1}{4} \cdot \frac{1 + {(- 1)}^{9} \cdot - 1 \frac{1^{9}}{4^{9}}}{1 - - \frac{1}{4}}$

Step 5.1.2.2

Multiply ${(- 1)}^{9}$ by $- 1$ .

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Step 5.1.2.2.1

Raise $- 1$ to the power of $1$ .

$- \frac{1}{4} \cdot \frac{1 + {(- 1)}^{9} \cdot {(- 1)}^{1} \frac{1^{9}}{4^{9}}}{1 - - \frac{1}{4}}$

Step 5.1.2.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- \frac{1}{4} \cdot \frac{1 + {(- 1)}^{9 + 1} \frac{1^{9}}{4^{9}}}{1 - - \frac{1}{4}}$

Step 5.1.2.3

Add $9$ and $1$ .

$- \frac{1}{4} \cdot \frac{1 + {(- 1)}^{10} \frac{1^{9}}{4^{9}}}{1 - - \frac{1}{4}}$

Step 5.1.3

Raise $- 1$ to the power of $10$ .

$- \frac{1}{4} \cdot \frac{1 + 1 \frac{1^{9}}{4^{9}}}{1 - - \frac{1}{4}}$

Step 5.1.4

Multiply $\frac{1^{9}}{4^{9}}$ by $1$ .

$- \frac{1}{4} \cdot \frac{1 + \frac{1^{9}}{4^{9}}}{1 - - \frac{1}{4}}$

Step 5.1.5

One to any power is one.

$- \frac{1}{4} \cdot \frac{1 + \frac{1}{4^{9}}}{1 - - \frac{1}{4}}$

Step 5.1.6

Raise $4$ to the power of $9$ .

$- \frac{1}{4} \cdot \frac{1 + \frac{1}{262144}}{1 - - \frac{1}{4}}$

Step 5.1.7

Write $1$ as a fraction with a common denominator.

$- \frac{1}{4} \cdot \frac{\frac{262144}{262144} + \frac{1}{262144}}{1 - - \frac{1}{4}}$

Step 5.1.8

Combine the numerators over the common denominator.

$- \frac{1}{4} \cdot \frac{\frac{262144 + 1}{262144}}{1 - - \frac{1}{4}}$

Step 5.1.9

Add $262144$ and $1$ .

$- \frac{1}{4} \cdot \frac{\frac{262145}{262144}}{1 - - \frac{1}{4}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- - \frac{1}{4}$ .

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Step 5.2.1.1

Multiply $- 1$ by $- 1$ .

$- \frac{1}{4} \cdot \frac{\frac{262145}{262144}}{1 + 1 (\frac{1}{4})}$

Step 5.2.1.2

Multiply $\frac{1}{4}$ by $1$ .

$- \frac{1}{4} \cdot \frac{\frac{262145}{262144}}{1 + \frac{1}{4}}$

Step 5.2.2

Write $1$ as a fraction with a common denominator.

$- \frac{1}{4} \cdot \frac{\frac{262145}{262144}}{\frac{4}{4} + \frac{1}{4}}$

Step 5.2.3

Combine the numerators over the common denominator.

$- \frac{1}{4} \cdot \frac{\frac{262145}{262144}}{\frac{4 + 1}{4}}$

Step 5.2.4

Add $4$ and $1$ .

$- \frac{1}{4} \cdot \frac{\frac{262145}{262144}}{\frac{5}{4}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

$- \frac{1}{4} (\frac{262145}{262144} \cdot \frac{4}{5})$

Step 5.4

Cancel the common factor of $5$ .

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Step 5.4.1

Factor $5$ out of $262145$ .

$- \frac{1}{4} (\frac{5 (52429)}{262144} \cdot \frac{4}{5})$

Step 5.4.2

Cancel the common factor.

$- \frac{1}{4} (\frac{5 \cdot 52429}{262144} \cdot \frac{4}{5})$

Step 5.4.3

Rewrite the expression.

$- \frac{1}{4} (\frac{52429}{262144} \cdot 4)$

Step 5.5

Cancel the common factor of $4$ .

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Step 5.5.1

Factor $4$ out of $262144$ .

$- \frac{1}{4} (\frac{52429}{4 (65536)} \cdot 4)$

Step 5.5.2

Cancel the common factor.

$- \frac{1}{4} (\frac{52429}{4 \cdot 65536} \cdot 4)$

Step 5.5.3

Rewrite the expression.

$- \frac{1}{4} \cdot \frac{52429}{65536}$

Step 5.6

Multiply $- \frac{1}{4} \cdot \frac{52429}{65536}$ .

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Step 5.6.1

Multiply $\frac{52429}{65536}$ by $\frac{1}{4}$ .

$- \frac{52429}{65536 \cdot 4}$

Step 5.6.2

Multiply $65536$ by $4$ .

$- \frac{52429}{262144}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{52429}{262144}$

Decimal Form:

$- 0.20000076 \dots$