Calculus Examples

$\sum_{x = 1}^{8} 15 {(- \frac{5}{7})}^{x - 1}$

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

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Step 2.1

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

$r = \frac{15 {(- \frac{5}{7})}^{x + 1 - 1}}{15 {(- \frac{5}{7})}^{x - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $15$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{15 {(- \frac{5}{7})}^{x + 1 - 1}}{15 {(- \frac{5}{7})}^{x - 1}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{(- \frac{5}{7})}^{x + 1 - 1}}{{(- \frac{5}{7})}^{x - 1}}$

Step 2.2.2

Cancel the common factor of ${(- \frac{5}{7})}^{x + 1 - 1}$ and ${(- \frac{5}{7})}^{x - 1}$ .

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Step 2.2.2.1

Factor ${(- \frac{5}{7})}^{x - 1}$ out of ${(- \frac{5}{7})}^{x + 1 - 1}$ .

$r = \frac{{(- \frac{5}{7})}^{x - 1} {(- \frac{5}{7})}^{x + 0 - (x - 1)}}{{(- \frac{5}{7})}^{x - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(- \frac{5}{7})}^{x - 1} {(- \frac{5}{7})}^{x + 0 - (x - 1)}}{{(- \frac{5}{7})}^{x - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(- \frac{5}{7})}^{x - 1} {(- \frac{5}{7})}^{x + 0 - (x - 1)}}{{(- \frac{5}{7})}^{x - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(- \frac{5}{7})}^{x + 0 - (x - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(- \frac{5}{7})}^{x + 0 - (x - 1)}$ by $1$ .

$r = {(- \frac{5}{7})}^{x + 0 - (x - 1)}$

Step 2.2.3

Add $x$ and $0$ .

$r = {(- \frac{5}{7})}^{x - (x - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {(- \frac{5}{7})}^{x - x - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = {(- \frac{5}{7})}^{x - x + 1}$

Step 2.2.5

Subtract $x$ from $x$ .

$r = {(- \frac{5}{7})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

$r = {(- \frac{5}{7})}^{1}$

Step 2.2.7

Simplify.

$r = - \frac{5}{7}$

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 $x$ into $15 {(- \frac{5}{7})}^{x - 1}$ .

$a = 15 {(- \frac{5}{7})}^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = 15 {(- \frac{5}{7})}^{0}$

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 $- \frac{5}{7}$ .

$a = 15 ({(- 1)}^{0} {(\frac{5}{7})}^{0})$

Step 3.2.2.2

Apply the product rule to $\frac{5}{7}$ .

$a = 15 ({(- 1)}^{0} \frac{5^{0}}{7^{0}})$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = 15 (1 \frac{5^{0}}{7^{0}})$

Step 3.2.4

Multiply $\frac{5^{0}}{7^{0}}$ by $1$ .

$a = 15 \frac{5^{0}}{7^{0}}$

Step 3.2.5

Anything raised to $0$ is $1$ .

$a = 15 \frac{1}{7^{0}}$

Step 3.2.6

Anything raised to $0$ is $1$ .

$a = 15 (\frac{1}{1})$

Step 3.2.7

Cancel the common factor of $1$ .

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Step 3.2.7.1

Cancel the common factor.

$a = 15 (\frac{1}{1})$

Step 3.2.7.2

Rewrite the expression.

$a = 15 \cdot 1$

Step 3.2.8

Multiply $15$ by $1$ .

$a = 15$

Step 4

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

$15 \frac{1 - {(- \frac{5}{7})}^{8}}{1 - - \frac{5}{7}}$

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 $- \frac{5}{7}$ .

$15 \frac{1 - ({(- 1)}^{8} {(\frac{5}{7})}^{8})}{1 - - \frac{5}{7}}$

Step 5.1.1.2

Apply the product rule to $\frac{5}{7}$ .

$15 \frac{1 - ({(- 1)}^{8} \frac{5^{8}}{7^{8}})}{1 - - \frac{5}{7}}$

Step 5.1.2

Multiply $- 1$ by ${(- 1)}^{8}$ by adding the exponents.

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Step 5.1.2.1

Move ${(- 1)}^{8}$ .

$15 \frac{1 + {(- 1)}^{8} \cdot - 1 \frac{5^{8}}{7^{8}}}{1 - - \frac{5}{7}}$

Step 5.1.2.2

Multiply ${(- 1)}^{8}$ by $- 1$ .

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Step 5.1.2.2.1

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

$15 \frac{1 + {(- 1)}^{8} \cdot {(- 1)}^{1} \frac{5^{8}}{7^{8}}}{1 - - \frac{5}{7}}$

Step 5.1.2.2.2

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

$15 \frac{1 + {(- 1)}^{8 + 1} \frac{5^{8}}{7^{8}}}{1 - - \frac{5}{7}}$

Step 5.1.2.3

Add $8$ and $1$ .

$15 \frac{1 + {(- 1)}^{9} \frac{5^{8}}{7^{8}}}{1 - - \frac{5}{7}}$

Step 5.1.3

Raise $- 1$ to the power of $9$ .

$15 \frac{1 - \frac{5^{8}}{7^{8}}}{1 - - \frac{5}{7}}$

Step 5.1.4

Raise $5$ to the power of $8$ .

$15 \frac{1 - \frac{390625}{7^{8}}}{1 - - \frac{5}{7}}$

Step 5.1.5

Raise $7$ to the power of $8$ .

$15 \frac{1 - \frac{390625}{5764801}}{1 - - \frac{5}{7}}$

Step 5.1.6

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

$15 \frac{\frac{5764801}{5764801} - \frac{390625}{5764801}}{1 - - \frac{5}{7}}$

Step 5.1.7

Combine the numerators over the common denominator.

$15 \frac{\frac{5764801 - 390625}{5764801}}{1 - - \frac{5}{7}}$

Step 5.1.8

Subtract $390625$ from $5764801$ .

$15 \frac{\frac{5374176}{5764801}}{1 - - \frac{5}{7}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- - \frac{5}{7}$ .

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Step 5.2.1.1

Multiply $- 1$ by $- 1$ .

$15 \frac{\frac{5374176}{5764801}}{1 + 1 (\frac{5}{7})}$

Step 5.2.1.2

Multiply $\frac{5}{7}$ by $1$ .

$15 \frac{\frac{5374176}{5764801}}{1 + \frac{5}{7}}$

Step 5.2.2

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

$15 \frac{\frac{5374176}{5764801}}{\frac{7}{7} + \frac{5}{7}}$

Step 5.2.3

Combine the numerators over the common denominator.

$15 \frac{\frac{5374176}{5764801}}{\frac{7 + 5}{7}}$

Step 5.2.4

Add $7$ and $5$ .

$15 \frac{\frac{5374176}{5764801}}{\frac{12}{7}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

$15 (\frac{5374176}{5764801} \cdot \frac{7}{12})$

Step 5.4

Cancel the common factor of $12$ .

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Step 5.4.1

Factor $12$ out of $5374176$ .

$15 (\frac{12 (447848)}{5764801} \cdot \frac{7}{12})$

Step 5.4.2

Cancel the common factor.

$15 (\frac{12 \cdot 447848}{5764801} \cdot \frac{7}{12})$

Step 5.4.3

Rewrite the expression.

$15 (\frac{447848}{5764801} \cdot 7)$

Step 5.5

Cancel the common factor of $7$ .

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Step 5.5.1

Factor $7$ out of $5764801$ .

$15 (\frac{447848}{7 (823543)} \cdot 7)$

Step 5.5.2

Cancel the common factor.

$15 (\frac{447848}{7 \cdot 823543} \cdot 7)$

Step 5.5.3

Rewrite the expression.

$15 (\frac{447848}{823543})$

Step 5.6

Multiply $15 (\frac{447848}{823543})$ .

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Step 5.6.1

Combine $15$ and $\frac{447848}{823543}$ .

$\frac{15 \cdot 447848}{823543}$

Step 5.6.2

Multiply $15$ by $447848$ .

$\frac{6717720}{823543}$