Precalculus 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

Substitute in the value of $n$ to find the $n$ th term.

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

Step 9

Simplify the numerator.

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Step 9.1

Subtract $1$ from $5$ .

$a_{5} = \frac{30 {(- 1)}^{4}}{3^{5 - 1}}$

Step 9.2

Raise $- 1$ to the power of $4$ .

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

Step 10

Simplify the denominator.

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Step 10.1

Subtract $1$ from $5$ .

$a_{5} = \frac{30 \cdot 1}{3^{4}}$

Step 10.2

Raise $3$ to the power of $4$ .

$a_{5} = \frac{30 \cdot 1}{81}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Multiply $30$ by $1$ .

$a_{5} = \frac{30}{81}$

Step 11.2

Cancel the common factor of $30$ and $81$ .

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Step 11.2.1

Factor $3$ out of $30$ .

$a_{5} = \frac{3 (10)}{81}$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $3$ out of $81$ .

$a_{5} = \frac{3 \cdot 10}{3 \cdot 27}$

Step 11.2.2.2

Cancel the common factor.

$a_{5} = \frac{3 \cdot 10}{3 \cdot 27}$

Step 11.2.2.3

Rewrite the expression.

$a_{5} = \frac{10}{27}$