Calculus Examples

$\sum_{i = 1}^{9} - 28 {(- \frac{1}{2})}^{i - 1}$

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

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Step 2.1

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

$r = \frac{- 28 {(- \frac{1}{2})}^{i + 1 - 1}}{- 28 {(- \frac{1}{2})}^{i - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $- 28$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{- 28 {(- \frac{1}{2})}^{i + 1 - 1}}{- 28 {(- \frac{1}{2})}^{i - 1}}$

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.2.2.4

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

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

Step 2.2.3

Add $i$ and $0$ .

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

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

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

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.5

Subtract $i$ from $i$ .

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

Step 2.2.6

Add $0$ and $1$ .

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

Step 2.2.7

Simplify.

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

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 $1$ for $i$ into $- 28 {(- \frac{1}{2})}^{i - 1}$ .

$a = - 28 {(- \frac{1}{2})}^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = - 28 {(- \frac{1}{2})}^{0}$

Step 3.2.2

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

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Step 3.2.2.1

Apply the product rule to $- \frac{1}{2}$ .

$a = - 28 ({(- 1)}^{0} {(\frac{1}{2})}^{0})$

Step 3.2.2.2

Apply the product rule to $\frac{1}{2}$ .

$a = - 28 ({(- 1)}^{0} \frac{1^{0}}{2^{0}})$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = - 28 (1 \frac{1^{0}}{2^{0}})$

Step 3.2.4

Multiply $\frac{1^{0}}{2^{0}}$ by $1$ .

$a = - 28 \frac{1^{0}}{2^{0}}$

Step 3.2.5

Anything raised to $0$ is $1$ .

$a = - 28 \frac{1}{2^{0}}$

Step 3.2.6

Anything raised to $0$ is $1$ .

$a = - 28 (\frac{1}{1})$

Step 3.2.7

Cancel the common factor of $1$ .

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Step 3.2.7.1

Cancel the common factor.

$a = - 28 (\frac{1}{1})$

Step 3.2.7.2

Rewrite the expression.

$a = - 28 \cdot 1$

Step 3.2.8

Multiply $- 28$ by $1$ .

$a = - 28$

Step 4

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

$- 28 \frac{1 - {(- \frac{1}{2})}^{9}}{1 - - \frac{1}{2}}$

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 $- \frac{1}{2}$ .

$- 28 \frac{1 - ({(- 1)}^{9} {(\frac{1}{2})}^{9})}{1 - - \frac{1}{2}}$

Step 5.1.1.2

Apply the product rule to $\frac{1}{2}$ .

$- 28 \frac{1 - ({(- 1)}^{9} \frac{1^{9}}{2^{9}})}{1 - - \frac{1}{2}}$

Step 5.1.2

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

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Step 5.1.2.1

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

$- 28 \frac{1 + {(- 1)}^{9} \cdot - 1 \frac{1^{9}}{2^{9}}}{1 - - \frac{1}{2}}$

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 $1$ .

$- 28 \frac{1 + {(- 1)}^{9} \cdot {(- 1)}^{1} \frac{1^{9}}{2^{9}}}{1 - - \frac{1}{2}}$

Step 5.1.2.2.2

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

$- 28 \frac{1 + {(- 1)}^{9 + 1} \frac{1^{9}}{2^{9}}}{1 - - \frac{1}{2}}$

Step 5.1.2.3

Add $9$ and $1$ .

$- 28 \frac{1 + {(- 1)}^{10} \frac{1^{9}}{2^{9}}}{1 - - \frac{1}{2}}$

Step 5.1.3

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

$- 28 \frac{1 + 1 \frac{1^{9}}{2^{9}}}{1 - - \frac{1}{2}}$

Step 5.1.4

Multiply $\frac{1^{9}}{2^{9}}$ by $1$ .

$- 28 \frac{1 + \frac{1^{9}}{2^{9}}}{1 - - \frac{1}{2}}$

Step 5.1.5

One to any power is one.

$- 28 \frac{1 + \frac{1}{2^{9}}}{1 - - \frac{1}{2}}$

Step 5.1.6

Raise $2$ to the power of $9$ .

$- 28 \frac{1 + \frac{1}{512}}{1 - - \frac{1}{2}}$

Step 5.1.7

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

$- 28 \frac{\frac{512}{512} + \frac{1}{512}}{1 - - \frac{1}{2}}$

Step 5.1.8

Combine the numerators over the common denominator.

$- 28 \frac{\frac{512 + 1}{512}}{1 - - \frac{1}{2}}$

Step 5.1.9

Add $512$ and $1$ .

$- 28 \frac{\frac{513}{512}}{1 - - \frac{1}{2}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- - \frac{1}{2}$ .

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Step 5.2.1.1

Multiply $- 1$ by $- 1$ .

$- 28 \frac{\frac{513}{512}}{1 + 1 (\frac{1}{2})}$

Step 5.2.1.2

Multiply $\frac{1}{2}$ by $1$ .

$- 28 \frac{\frac{513}{512}}{1 + \frac{1}{2}}$

Step 5.2.2

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

$- 28 \frac{\frac{513}{512}}{\frac{2}{2} + \frac{1}{2}}$

Step 5.2.3

Combine the numerators over the common denominator.

$- 28 \frac{\frac{513}{512}}{\frac{2 + 1}{2}}$

Step 5.2.4

Add $2$ and $1$ .

$- 28 \frac{\frac{513}{512}}{\frac{3}{2}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

$- 28 (\frac{513}{512} \cdot \frac{2}{3})$

Step 5.4

Cancel the common factor of $3$ .

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Step 5.4.1

Factor $3$ out of $513$ .

$- 28 (\frac{3 (171)}{512} \cdot \frac{2}{3})$

Step 5.4.2

Cancel the common factor.

$- 28 (\frac{3 \cdot 171}{512} \cdot \frac{2}{3})$

Step 5.4.3

Rewrite the expression.

$- 28 (\frac{171}{512} \cdot 2)$

Step 5.5

Cancel the common factor of $2$ .

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Step 5.5.1

Factor $2$ out of $512$ .

$- 28 (\frac{171}{2 (256)} \cdot 2)$

Step 5.5.2

Cancel the common factor.

$- 28 (\frac{171}{2 \cdot 256} \cdot 2)$

Step 5.5.3

Rewrite the expression.

$- 28 (\frac{171}{256})$

Step 5.6

Cancel the common factor of $4$ .

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Step 5.6.1

Factor $4$ out of $- 28$ .

$4 (- 7) \frac{171}{256}$

Step 5.6.2

Factor $4$ out of $256$ .

$4 \cdot - 7 \frac{171}{4 \cdot 64}$

Step 5.6.3

Cancel the common factor.

$4 \cdot - 7 \frac{171}{4 \cdot 64}$

Step 5.6.4

Rewrite the expression.

$- 7 (\frac{171}{64})$

Step 5.7

Combine $- 7$ and $\frac{171}{64}$ .

$\frac{- 7 \cdot 171}{64}$

Step 5.8

Multiply $- 7$ by $171$ .

$\frac{- 1197}{64}$

Step 5.9

Move the negative in front of the fraction.

$- \frac{1197}{64}$