Precalculus Examples

$\frac{1}{3}$ , $- \frac{1}{9}$ , $\frac{1}{27}$ , $- \frac{1}{81}$

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 $a_{1} = \frac{1}{3}$ and $r = - \frac{1}{3}$ .

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

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

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

Step 4.2

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

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

Step 5

One to any power is one.

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

Step 6

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

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

Step 7

Combine.

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

Step 8

Multiply $3$ by $3^{n - 1}$ by adding the exponents.

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Step 8.1

Multiply $3$ by $3^{n - 1}$ .

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Step 8.1.1

Raise $3$ to the power of $1$ .

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

Step 8.1.2

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

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

Step 8.2

Combine the opposite terms in $1 + n - 1$ .

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Step 8.2.1

Subtract $1$ from $1$ .

$a_{n} = \frac{1 {(- 1)}^{n - 1}}{3^{n + 0}}$

Step 8.2.2

Add $n$ and $0$ .

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

Step 9

Multiply ${(- 1)}^{n - 1}$ by $1$ .

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