Calculus Examples

$\sum_{n = 1}^{4} (125) {(\frac{1}{5})}^{n - 1}$

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

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Step 2.1

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

$r = \frac{125 {(\frac{1}{5})}^{n + 1 - 1}}{125 {(\frac{1}{5})}^{n - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $125$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{125 {(\frac{1}{5})}^{n + 1 - 1}}{125 {(\frac{1}{5})}^{n - 1}}$

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.2.2.4

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

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

Step 2.2.3

Add $n$ and $0$ .

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

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

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

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.5

Subtract $n$ from $n$ .

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

Step 2.2.6

Add $0$ and $1$ .

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

Step 2.2.7

Simplify.

$r = \frac{1}{5}$

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 $n$ into $(125) {(\frac{1}{5})}^{n - 1}$ .

$a = 125 {(\frac{1}{5})}^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = 125 {(\frac{1}{5})}^{0}$

Step 3.2.2

Apply the product rule to $\frac{1}{5}$ .

$a = 125 \frac{1^{0}}{5^{0}}$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = 125 \frac{1}{5^{0}}$

Step 3.2.4

Anything raised to $0$ is $1$ .

$a = 125 (\frac{1}{1})$

Step 3.2.5

Cancel the common factor of $1$ .

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Step 3.2.5.1

Cancel the common factor.

$a = 125 (\frac{1}{1})$

Step 3.2.5.2

Rewrite the expression.

$a = 125 \cdot 1$

Step 3.2.6

Multiply $125$ by $1$ .

$a = 125$

Step 4

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

$125 \frac{1 - {(\frac{1}{5})}^{4}}{1 - \frac{1}{5}}$

Step 5

Simplify.

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Step 5.1

Multiply the numerator and denominator of the fraction by $5$ .

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Step 5.1.1

Multiply $\frac{1 - {(\frac{1}{5})}^{4}}{1 - \frac{1}{5}}$ by $\frac{5}{5}$ .

$125 (\frac{5}{5} \cdot \frac{1 - {(\frac{1}{5})}^{4}}{1 - \frac{1}{5}})$

Step 5.1.2

Combine.

$125 \frac{5 (1 - {(\frac{1}{5})}^{4})}{5 (1 - \frac{1}{5})}$

Step 5.2

Apply the distributive property.

$125 \frac{5 \cdot 1 + 5 (- {(\frac{1}{5})}^{4})}{5 \cdot 1 + 5 (- \frac{1}{5})}$

Step 5.3

Cancel the common factor of $5$ .

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Step 5.3.1

Move the leading negative in $- \frac{1}{5}$ into the numerator.

$125 \frac{5 \cdot 1 + 5 (- {(\frac{1}{5})}^{4})}{5 \cdot 1 + 5 (\frac{- 1}{5})}$

Step 5.3.2

Cancel the common factor.

$125 \frac{5 \cdot 1 + 5 (- {(\frac{1}{5})}^{4})}{5 \cdot 1 + 5 (\frac{- 1}{5})}$

Step 5.3.3

Rewrite the expression.

$125 \frac{5 \cdot 1 + 5 (- {(\frac{1}{5})}^{4})}{5 \cdot 1 - 1}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $5$ by $1$ .

$125 \frac{5 + 5 (- {(\frac{1}{5})}^{4})}{5 \cdot 1 - 1}$

Step 5.4.2

Apply the product rule to $\frac{1}{5}$ .

$125 \frac{5 + 5 (- \frac{1^{4}}{5^{4}})}{5 \cdot 1 - 1}$

Step 5.4.3

Cancel the common factor of $5$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{1^{4}}{5^{4}}$ into the numerator.

$125 \frac{5 + 5 \frac{- 1^{4}}{5^{4}}}{5 \cdot 1 - 1}$

Step 5.4.3.2

Factor $5$ out of $5^{4}$ .

$125 \frac{5 + 5 \frac{- 1^{4}}{5 \cdot 5^{3}}}{5 \cdot 1 - 1}$

Step 5.4.3.3

Cancel the common factor.

$125 \frac{5 + 5 \frac{- 1^{4}}{5 \cdot 5^{3}}}{5 \cdot 1 - 1}$

Step 5.4.3.4

Rewrite the expression.

$125 \frac{5 + \frac{- 1^{4}}{5^{3}}}{5 \cdot 1 - 1}$

Step 5.4.4

One to any power is one.

$125 \frac{5 + \frac{- 1 \cdot 1}{5^{3}}}{5 \cdot 1 - 1}$

Step 5.4.5

Raise $5$ to the power of $3$ .

$125 \frac{5 + \frac{- 1 \cdot 1}{125}}{5 \cdot 1 - 1}$

Step 5.4.6

Multiply $- 1$ by $1$ .

$125 \frac{5 + \frac{- 1}{125}}{5 \cdot 1 - 1}$

Step 5.4.7

Move the negative in front of the fraction.

$125 \frac{5 - \frac{1}{125}}{5 \cdot 1 - 1}$

Step 5.4.8

To write $5$ as a fraction with a common denominator, multiply by $\frac{125}{125}$ .

$125 \frac{5 \cdot \frac{125}{125} - \frac{1}{125}}{5 \cdot 1 - 1}$

Step 5.4.9

Combine $5$ and $\frac{125}{125}$ .

$125 \frac{\frac{5 \cdot 125}{125} - \frac{1}{125}}{5 \cdot 1 - 1}$

Step 5.4.10

Combine the numerators over the common denominator.

$125 \frac{\frac{5 \cdot 125 - 1}{125}}{5 \cdot 1 - 1}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $5$ by $125$ .

$125 \frac{\frac{625 - 1}{125}}{5 \cdot 1 - 1}$

Step 5.4.11.2

Subtract $1$ from $625$ .

$125 \frac{\frac{624}{125}}{5 \cdot 1 - 1}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $5$ by $1$ .

$125 \frac{\frac{624}{125}}{5 - 1}$

Step 5.5.2

Subtract $1$ from $5$ .

$125 \frac{\frac{624}{125}}{4}$

Step 5.6

Multiply the numerator by the reciprocal of the denominator.

$125 (\frac{624}{125} \cdot \frac{1}{4})$

Step 5.7

Cancel the common factor of $4$ .

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Step 5.7.1

Factor $4$ out of $624$ .

$125 (\frac{4 (156)}{125} \cdot \frac{1}{4})$

Step 5.7.2

Cancel the common factor.

$125 (\frac{4 \cdot 156}{125} \cdot \frac{1}{4})$

Step 5.7.3

Rewrite the expression.

$125 (\frac{156}{125})$

Step 5.8

Cancel the common factor of $125$ .

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Step 5.8.1

Cancel the common factor.

$125 (\frac{156}{125})$

Step 5.8.2

Rewrite the expression.

$156$