Precalculus Examples

$\sum_{i = 1}^{6} 12 {(\frac{5}{6})}^{i - 1}$

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 $r$ .

$r = \frac{12 {(\frac{5}{6})}^{i + 1 - 1}}{12 {(\frac{5}{6})}^{i - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $12$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{12 {(\frac{5}{6})}^{i + 1 - 1}}{12 {(\frac{5}{6})}^{i - 1}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{{(\frac{5}{6})}^{i + 1 - 1}}{{(\frac{5}{6})}^{i - 1}}$

Step 2.2.2

Cancel the common factor of ${(\frac{5}{6})}^{i + 1 - 1}$ and ${(\frac{5}{6})}^{i - 1}$ .

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Step 2.2.2.1

Factor ${(\frac{5}{6})}^{i - 1}$ out of ${(\frac{5}{6})}^{i + 1 - 1}$ .

$r = \frac{{(\frac{5}{6})}^{i - 1} {(\frac{5}{6})}^{i + 0 - (i - 1)}}{{(\frac{5}{6})}^{i - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{{(\frac{5}{6})}^{i - 1} {(\frac{5}{6})}^{i + 0 - (i - 1)}}{{(\frac{5}{6})}^{i - 1} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{{(\frac{5}{6})}^{i - 1} {(\frac{5}{6})}^{i + 0 - (i - 1)}}{{(\frac{5}{6})}^{i - 1} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(\frac{5}{6})}^{i + 0 - (i - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(\frac{5}{6})}^{i + 0 - (i - 1)}$ by $1$ .

$r = {(\frac{5}{6})}^{i + 0 - (i - 1)}$

Step 2.2.3

Add $i$ and $0$ .

$r = {(\frac{5}{6})}^{i - (i - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {(\frac{5}{6})}^{i - i - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = {(\frac{5}{6})}^{i - i + 1}$

Step 2.2.5

Subtract $i$ from $i$ .

$r = {(\frac{5}{6})}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

$r = {(\frac{5}{6})}^{1}$

Step 2.2.7

Simplify.

$r = \frac{5}{6}$

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 $12 {(\frac{5}{6})}^{i - 1}$ .

$a = 12 {(\frac{5}{6})}^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = 12 {(\frac{5}{6})}^{0}$

Step 3.2.2

Apply the product rule to $\frac{5}{6}$ .

$a = 12 \frac{5^{0}}{6^{0}}$

Step 3.2.3

Anything raised to $0$ is $1$ .

$a = 12 \frac{1}{6^{0}}$

Step 3.2.4

Anything raised to $0$ is $1$ .

$a = 12 (\frac{1}{1})$

Step 3.2.5

Cancel the common factor of $1$ .

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Step 3.2.5.1

Cancel the common factor.

$a = 12 (\frac{1}{1})$

Step 3.2.5.2

Rewrite the expression.

$a = 12 \cdot 1$

Step 3.2.6

Multiply $12$ by $1$ .

$a = 12$

Step 4

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

$12 \frac{1 - {(\frac{5}{6})}^{6}}{1 - \frac{5}{6}}$

Step 5

Simplify.

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Step 5.1

Multiply the numerator and denominator of the fraction by $6$ .

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Step 5.1.1

Multiply $\frac{1 - {(\frac{5}{6})}^{6}}{1 - \frac{5}{6}}$ by $\frac{6}{6}$ .

$12 (\frac{6}{6} \cdot \frac{1 - {(\frac{5}{6})}^{6}}{1 - \frac{5}{6}})$

Step 5.1.2

Combine.

$12 \frac{6 (1 - {(\frac{5}{6})}^{6})}{6 (1 - \frac{5}{6})}$

Step 5.2

Apply the distributive property.

$12 \frac{6 \cdot 1 + 6 (- {(\frac{5}{6})}^{6})}{6 \cdot 1 + 6 (- \frac{5}{6})}$

Step 5.3

Cancel the common factor of $6$ .

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Step 5.3.1

Move the leading negative in $- \frac{5}{6}$ into the numerator.

$12 \frac{6 \cdot 1 + 6 (- {(\frac{5}{6})}^{6})}{6 \cdot 1 + 6 (\frac{- 5}{6})}$

Step 5.3.2

Cancel the common factor.

$12 \frac{6 \cdot 1 + 6 (- {(\frac{5}{6})}^{6})}{6 \cdot 1 + 6 (\frac{- 5}{6})}$

Step 5.3.3

Rewrite the expression.

$12 \frac{6 \cdot 1 + 6 (- {(\frac{5}{6})}^{6})}{6 \cdot 1 - 5}$

Step 5.4

Simplify the numerator.

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Step 5.4.1

Multiply $6$ by $1$ .

$12 \frac{6 + 6 (- {(\frac{5}{6})}^{6})}{6 \cdot 1 - 5}$

Step 5.4.2

Apply the product rule to $\frac{5}{6}$ .

$12 \frac{6 + 6 (- \frac{5^{6}}{6^{6}})}{6 \cdot 1 - 5}$

Step 5.4.3

Cancel the common factor of $6$ .

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Step 5.4.3.1

Move the leading negative in $- \frac{5^{6}}{6^{6}}$ into the numerator.

$12 \frac{6 + 6 \frac{- 5^{6}}{6^{6}}}{6 \cdot 1 - 5}$

Step 5.4.3.2

Factor $6$ out of $6^{6}$ .

$12 \frac{6 + 6 \frac{- 5^{6}}{6 \cdot 6^{5}}}{6 \cdot 1 - 5}$

Step 5.4.3.3

Cancel the common factor.

$12 \frac{6 + 6 \frac{- 5^{6}}{6 \cdot 6^{5}}}{6 \cdot 1 - 5}$

Step 5.4.3.4

Rewrite the expression.

$12 \frac{6 + \frac{- 5^{6}}{6^{5}}}{6 \cdot 1 - 5}$

Step 5.4.4

Raise $5$ to the power of $6$ .

$12 \frac{6 + \frac{- 1 \cdot 15625}{6^{5}}}{6 \cdot 1 - 5}$

Step 5.4.5

Raise $6$ to the power of $5$ .

$12 \frac{6 + \frac{- 1 \cdot 15625}{7776}}{6 \cdot 1 - 5}$

Step 5.4.6

Multiply $- 1$ by $15625$ .

$12 \frac{6 + \frac{- 15625}{7776}}{6 \cdot 1 - 5}$

Step 5.4.7

Move the negative in front of the fraction.

$12 \frac{6 - \frac{15625}{7776}}{6 \cdot 1 - 5}$

Step 5.4.8

To write $6$ as a fraction with a common denominator, multiply by $\frac{7776}{7776}$ .

$12 \frac{6 \cdot \frac{7776}{7776} - \frac{15625}{7776}}{6 \cdot 1 - 5}$

Step 5.4.9

Combine $6$ and $\frac{7776}{7776}$ .

$12 \frac{\frac{6 \cdot 7776}{7776} - \frac{15625}{7776}}{6 \cdot 1 - 5}$

Step 5.4.10

Combine the numerators over the common denominator.

$12 \frac{\frac{6 \cdot 7776 - 15625}{7776}}{6 \cdot 1 - 5}$

Step 5.4.11

Simplify the numerator.

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Step 5.4.11.1

Multiply $6$ by $7776$ .

$12 \frac{\frac{46656 - 15625}{7776}}{6 \cdot 1 - 5}$

Step 5.4.11.2

Subtract $15625$ from $46656$ .

$12 \frac{\frac{31031}{7776}}{6 \cdot 1 - 5}$

Step 5.5

Simplify the denominator.

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Step 5.5.1

Multiply $6$ by $1$ .

$12 \frac{\frac{31031}{7776}}{6 - 5}$

Step 5.5.2

Subtract $5$ from $6$ .

$12 \frac{\frac{31031}{7776}}{1}$

Step 5.6

Divide $\frac{31031}{7776}$ by $1$ .

$12 (\frac{31031}{7776})$

Step 5.7

Cancel the common factor of $12$ .

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Step 5.7.1

Factor $12$ out of $7776$ .

$12 \frac{31031}{12 (648)}$

Step 5.7.2

Cancel the common factor.

$12 \frac{31031}{12 \cdot 648}$

Step 5.7.3

Rewrite the expression.

$\frac{31031}{648}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{31031}{648}$

Decimal Form:

$47.88734567 \dots$

Mixed Number Form:

$47 \frac{575}{648}$