Precalculus Examples

$\sum_{k = 1}^{8} - 3 {(- \frac{3}{5})}^{k - 1}$

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

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Step 2.1

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

$r = \frac{- 3 {(- \frac{3}{5})}^{k + 1 - 1}}{- 3 {(- \frac{3}{5})}^{k - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $- 3$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{- 3 {(- \frac{3}{5})}^{k + 1 - 1}}{- 3 {(- \frac{3}{5})}^{k - 1}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{(- \frac{3}{5})}^{k + 1 - 1}}{{(- \frac{3}{5})}^{k - 1}}$

Step 2.2.2

Cancel the common factor of ${(- \frac{3}{5})}^{k + 1 - 1}$ and ${(- \frac{3}{5})}^{k - 1}$ .

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Step 2.2.2.1

Factor ${(- \frac{3}{5})}^{k - 1}$ out of ${(- \frac{3}{5})}^{k + 1 - 1}$ .

$r = \frac{{(- \frac{3}{5})}^{k - 1} {(- \frac{3}{5})}^{k + 0 - (k - 1)}}{{(- \frac{3}{5})}^{k - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(- \frac{3}{5})}^{k - 1} {(- \frac{3}{5})}^{k + 0 - (k - 1)}}{{(- \frac{3}{5})}^{k - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(- \frac{3}{5})}^{k - 1} {(- \frac{3}{5})}^{k + 0 - (k - 1)}}{{(- \frac{3}{5})}^{k - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(- \frac{3}{5})}^{k + 0 - (k - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(- \frac{3}{5})}^{k + 0 - (k - 1)}$ by $1$ .

$r = {(- \frac{3}{5})}^{k + 0 - (k - 1)}$

Step 2.2.3

Add $k$ and $0$ .

$r = {(- \frac{3}{5})}^{k - (k - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {(- \frac{3}{5})}^{k - k - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = {(- \frac{3}{5})}^{k - k + 1}$

Step 2.2.5

Subtract $k$ from $k$ .

$r = {(- \frac{3}{5})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

$r = {(- \frac{3}{5})}^{1}$

Step 2.2.7

Simplify.

$r = - \frac{3}{5}$

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 $k$ into $- 3 {(- \frac{3}{5})}^{k - 1}$ .

$a = - 3 {(- \frac{3}{5})}^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = - 3 {(- \frac{3}{5})}^{0}$

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 $- \frac{3}{5}$ .

$a = - 3 ({(- 1)}^{0} {(\frac{3}{5})}^{0})$

Step 3.2.2.2

Apply the product rule to $\frac{3}{5}$ .

$a = - 3 ({(- 1)}^{0} \frac{3^{0}}{5^{0}})$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = - 3 (1 \frac{3^{0}}{5^{0}})$

Step 3.2.4

Multiply $\frac{3^{0}}{5^{0}}$ by $1$ .

$a = - 3 \frac{3^{0}}{5^{0}}$

Step 3.2.5

Anything raised to $0$ is $1$ .

$a = - 3 \frac{1}{5^{0}}$

Step 3.2.6

Anything raised to $0$ is $1$ .

$a = - 3 (\frac{1}{1})$

Step 3.2.7

Cancel the common factor of $1$ .

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Step 3.2.7.1

Cancel the common factor.

$a = - 3 (\frac{1}{1})$

Step 3.2.7.2

Rewrite the expression.

$a = - 3 \cdot 1$

Step 3.2.8

Multiply $- 3$ by $1$ .

$a = - 3$

Step 4

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

$- 3 \frac{1 - {(- \frac{3}{5})}^{8}}{1 - - \frac{3}{5}}$

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 $- \frac{3}{5}$ .

$- 3 \frac{1 - ({(- 1)}^{8} {(\frac{3}{5})}^{8})}{1 - - \frac{3}{5}}$

Step 5.1.1.2

Apply the product rule to $\frac{3}{5}$ .

$- 3 \frac{1 - ({(- 1)}^{8} \frac{3^{8}}{5^{8}})}{1 - - \frac{3}{5}}$

Step 5.1.2

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

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Step 5.1.2.1

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

$- 3 \frac{1 + {(- 1)}^{8} \cdot - 1 \frac{3^{8}}{5^{8}}}{1 - - \frac{3}{5}}$

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 $1$ .

$- 3 \frac{1 + {(- 1)}^{8} \cdot {(- 1)}^{1} \frac{3^{8}}{5^{8}}}{1 - - \frac{3}{5}}$

Step 5.1.2.2.2

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

$- 3 \frac{1 + {(- 1)}^{8 + 1} \frac{3^{8}}{5^{8}}}{1 - - \frac{3}{5}}$

Step 5.1.2.3

Add $8$ and $1$ .

$- 3 \frac{1 + {(- 1)}^{9} \frac{3^{8}}{5^{8}}}{1 - - \frac{3}{5}}$

Step 5.1.3

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

$- 3 \frac{1 - \frac{3^{8}}{5^{8}}}{1 - - \frac{3}{5}}$

Step 5.1.4

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

$- 3 \frac{1 - \frac{6561}{5^{8}}}{1 - - \frac{3}{5}}$

Step 5.1.5

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

$- 3 \frac{1 - \frac{6561}{390625}}{1 - - \frac{3}{5}}$

Step 5.1.6

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

$- 3 \frac{\frac{390625}{390625} - \frac{6561}{390625}}{1 - - \frac{3}{5}}$

Step 5.1.7

Combine the numerators over the common denominator.

$- 3 \frac{\frac{390625 - 6561}{390625}}{1 - - \frac{3}{5}}$

Step 5.1.8

Subtract $6561$ from $390625$ .

$- 3 \frac{\frac{384064}{390625}}{1 - - \frac{3}{5}}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- - \frac{3}{5}$ .

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Step 5.2.1.1

Multiply $- 1$ by $- 1$ .

$- 3 \frac{\frac{384064}{390625}}{1 + 1 (\frac{3}{5})}$

Step 5.2.1.2

Multiply $\frac{3}{5}$ by $1$ .

$- 3 \frac{\frac{384064}{390625}}{1 + \frac{3}{5}}$

Step 5.2.2

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

$- 3 \frac{\frac{384064}{390625}}{\frac{5}{5} + \frac{3}{5}}$

Step 5.2.3

Combine the numerators over the common denominator.

$- 3 \frac{\frac{384064}{390625}}{\frac{5 + 3}{5}}$

Step 5.2.4

Add $5$ and $3$ .

$- 3 \frac{\frac{384064}{390625}}{\frac{8}{5}}$

Step 5.3

Multiply the numerator by the reciprocal of the denominator.

$- 3 (\frac{384064}{390625} \cdot \frac{5}{8})$

Step 5.4

Cancel the common factor of $8$ .

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Step 5.4.1

Factor $8$ out of $384064$ .

$- 3 (\frac{8 (48008)}{390625} \cdot \frac{5}{8})$

Step 5.4.2

Cancel the common factor.

$- 3 (\frac{8 \cdot 48008}{390625} \cdot \frac{5}{8})$

Step 5.4.3

Rewrite the expression.

$- 3 (\frac{48008}{390625} \cdot 5)$

Step 5.5

Cancel the common factor of $5$ .

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Step 5.5.1

Factor $5$ out of $390625$ .

$- 3 (\frac{48008}{5 (78125)} \cdot 5)$

Step 5.5.2

Cancel the common factor.

$- 3 (\frac{48008}{5 \cdot 78125} \cdot 5)$

Step 5.5.3

Rewrite the expression.

$- 3 (\frac{48008}{78125})$

Step 5.6

Multiply $- 3 (\frac{48008}{78125})$ .

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Step 5.6.1

Combine $- 3$ and $\frac{48008}{78125}$ .

$\frac{- 3 \cdot 48008}{78125}$

Step 5.6.2

Multiply $- 3$ by $48008$ .

$\frac{- 144024}{78125}$

Step 5.7

Move the negative in front of the fraction.

$- \frac{144024}{78125}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{144024}{78125}$

Decimal Form:

$- 1.8435072$

Mixed Number Form:

$- 1 \frac{65899}{78125}$