Algebra Examples

$30$ , $- 10$ , $\frac{10}{3}$ , $- \frac{10}{9}$

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $- \frac{1}{3}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = - \frac{1}{3}$

Step 2

This is the form of a geometric sequence.

$a_{n} = a_{1} r^{n - 1}$

Step 3

Substitute in the values of $a_{1} = 30$ and $r = - \frac{1}{3}$ .

$a_{n} = 30 {(- \frac{1}{3})}^{n - 1}$

Step 4

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

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Step 4.1

Apply the product rule to $- \frac{1}{3}$ .

$a_{n} = 30 ({(- 1)}^{n - 1} {(\frac{1}{3})}^{n - 1})$

Step 4.2

Apply the product rule to $\frac{1}{3}$ .

$a_{n} = 30 ({(- 1)}^{n - 1} \frac{1^{n - 1}}{3^{n - 1}})$

Step 5

One to any power is one.

$a_{n} = 30 ({(- 1)}^{n - 1} \frac{1}{3^{n - 1}})$

Step 6

Combine ${(- 1)}^{n - 1}$ and $\frac{1}{3^{n - 1}}$ .

$a_{n} = 30 \frac{{(- 1)}^{n - 1}}{3^{n - 1}}$

Step 7

Combine $30$ and $\frac{{(- 1)}^{n - 1}}{3^{n - 1}}$ .

$a_{n} = \frac{30 {(- 1)}^{n - 1}}{3^{n - 1}}$

Step 8

This is the formula to find the sum of the first $n$ terms of the geometric sequence. To evaluate it, find the values of $r$ and $a_{1}$ .

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

Step 9

Replace the variables with the known values to find $S_{5}$ .

$S_{5} = 30 \cdot \frac{{(- \frac{1}{3})}^{5} - 1}{- \frac{1}{3} - 1}$

Step 10

Simplify the numerator.

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Step 10.1

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

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Step 10.1.1

Apply the product rule to $- \frac{1}{3}$ .

$S_{5} = 30 \cdot \frac{{(- 1)}^{5} {(\frac{1}{3})}^{5} - 1}{- \frac{1}{3} - 1}$

Step 10.1.2

Apply the product rule to $\frac{1}{3}$ .

$S_{5} = 30 \cdot \frac{{(- 1)}^{5} (\frac{1^{5}}{3^{5}}) - 1}{- \frac{1}{3} - 1}$

Step 10.2

Raise $- 1$ to the power of $5$ .

$S_{5} = 30 \cdot \frac{- \frac{1^{5}}{3^{5}} - 1}{- \frac{1}{3} - 1}$

Step 10.3

One to any power is one.

$S_{5} = 30 \cdot \frac{- \frac{1}{3^{5}} - 1}{- \frac{1}{3} - 1}$

Step 10.4

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

$S_{5} = 30 \cdot \frac{- \frac{1}{243} - 1}{- \frac{1}{3} - 1}$

Step 10.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{243}{243}$ .

$S_{5} = 30 \cdot \frac{- \frac{1}{243} - 1 \cdot \frac{243}{243}}{- \frac{1}{3} - 1}$

Step 10.6

Combine $- 1$ and $\frac{243}{243}$ .

$S_{5} = 30 \cdot \frac{- \frac{1}{243} + \frac{- 1 \cdot 243}{243}}{- \frac{1}{3} - 1}$

Step 10.7

Combine the numerators over the common denominator.

$S_{5} = 30 \cdot \frac{\frac{- 1 - 1 \cdot 243}{243}}{- \frac{1}{3} - 1}$

Step 10.8

Simplify the numerator.

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Step 10.8.1

Multiply $- 1$ by $243$ .

$S_{5} = 30 \cdot \frac{\frac{- 1 - 243}{243}}{- \frac{1}{3} - 1}$

Step 10.8.2

Subtract $243$ from $- 1$ .

$S_{5} = 30 \cdot \frac{\frac{- 244}{243}}{- \frac{1}{3} - 1}$

Step 10.9

Move the negative in front of the fraction.

$S_{5} = 30 \cdot \frac{- \frac{244}{243}}{- \frac{1}{3} - 1}$

Step 11

Simplify the denominator.

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Step 11.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$S_{5} = 30 \cdot \frac{- \frac{244}{243}}{- \frac{1}{3} - 1 \cdot \frac{3}{3}}$

Step 11.2

Combine $- 1$ and $\frac{3}{3}$ .

$S_{5} = 30 \cdot \frac{- \frac{244}{243}}{- \frac{1}{3} + \frac{- 1 \cdot 3}{3}}$

Step 11.3

Combine the numerators over the common denominator.

$S_{5} = 30 \cdot \frac{- \frac{244}{243}}{\frac{- 1 - 1 \cdot 3}{3}}$

Step 11.4

Simplify the numerator.

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Step 11.4.1

Multiply $- 1$ by $3$ .

$S_{5} = 30 \cdot \frac{- \frac{244}{243}}{\frac{- 1 - 3}{3}}$

Step 11.4.2

Subtract $3$ from $- 1$ .

$S_{5} = 30 \cdot \frac{- \frac{244}{243}}{\frac{- 4}{3}}$

Step 11.5

Move the negative in front of the fraction.

$S_{5} = 30 \cdot \frac{- \frac{244}{243}}{- \frac{4}{3}}$

Step 12

Dividing two negative values results in a positive value.

$S_{5} = 30 \cdot \frac{\frac{244}{243}}{\frac{4}{3}}$

Step 13

Multiply the numerator by the reciprocal of the denominator.

$S_{5} = 30 \cdot (\frac{244}{243} \cdot \frac{3}{4})$

Step 14

Cancel the common factor of $4$ .

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Step 14.1

Factor $4$ out of $244$ .

$S_{5} = 30 \cdot (\frac{4 (61)}{243} \cdot \frac{3}{4})$

Step 14.2

Cancel the common factor.

$S_{5} = 30 \cdot (\frac{4 \cdot 61}{243} \cdot \frac{3}{4})$

Step 14.3

Rewrite the expression.

$S_{5} = 30 \cdot (\frac{61}{243} \cdot 3)$

Step 15

Cancel the common factor of $3$ .

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Step 15.1

Factor $3$ out of $243$ .

$S_{5} = 30 \cdot (\frac{61}{3 (81)} \cdot 3)$

Step 15.2

Cancel the common factor.

$S_{5} = 30 \cdot (\frac{61}{3 \cdot 81} \cdot 3)$

Step 15.3

Rewrite the expression.

$S_{5} = 30 \cdot \frac{61}{81}$

Step 16

Cancel the common factor of $3$ .

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Step 16.1

Factor $3$ out of $30$ .

$S_{5} = 3 (10) \cdot \frac{61}{81}$

Step 16.2

Factor $3$ out of $81$ .

$S_{5} = 3 \cdot 10 \cdot \frac{61}{3 \cdot 27}$

Step 16.3

Cancel the common factor.

$S_{5} = 3 \cdot 10 \cdot \frac{61}{3 \cdot 27}$

Step 16.4

Rewrite the expression.

$S_{5} = 10 \cdot \frac{61}{27}$

Step 17

Combine $10$ and $\frac{61}{27}$ .

$S_{5} = \frac{10 \cdot 61}{27}$

Step 18

Multiply $10$ by $61$ .

$S_{5} = \frac{610}{27}$