Precalculus Examples

$- \frac{1}{3}$ , $\frac{1}{6}$ , $- \frac{1}{12}$

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 $- \frac{1}{2}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = - \frac{1}{2}$

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 $r = - \frac{1}{2}$ .

$a_{n} = - \frac{1}{3} {(- \frac{1}{2})}^{n - 1}$

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 $- \frac{1}{2}$ .

$a_{n} = - \frac{1}{3} ({(- 1)}^{n - 1} {(\frac{1}{2})}^{n - 1})$

Step 4.2

Apply the product rule to $\frac{1}{2}$ .

$a_{n} = - \frac{1}{3} ({(- 1)}^{n - 1} \frac{1^{n - 1}}{2^{n - 1}})$

Step 5

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

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Step 5.1

Move ${(- 1)}^{n - 1}$ .

$a_{n} = {(- 1)}^{n - 1} \cdot - 1 \frac{1}{3} \frac{1^{n - 1}}{2^{n - 1}}$

Step 5.2

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

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Step 5.2.1

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

$a_{n} = {(- 1)}^{n - 1} \cdot {(- 1)}^{1} \frac{1}{3} \frac{1^{n - 1}}{2^{n - 1}}$

Step 5.2.2

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

$a_{n} = {(- 1)}^{n - 1 + 1} \frac{1}{3} \frac{1^{n - 1}}{2^{n - 1}}$

Step 5.3

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

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Step 5.3.1

Add $- 1$ and $1$ .

$a_{n} = {(- 1)}^{n + 0} \frac{1}{3} \frac{1^{n - 1}}{2^{n - 1}}$

Step 5.3.2

Add $n$ and $0$ .

$a_{n} = {(- 1)}^{n} \frac{1}{3} \frac{1^{n - 1}}{2^{n - 1}}$

Step 6

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

$a_{n} = \frac{{(- 1)}^{n}}{3} \cdot \frac{1^{n - 1}}{2^{n - 1}}$

Step 7

Combine.

$a_{n} = \frac{{(- 1)}^{n} \cdot 1^{n - 1}}{3 \cdot 2^{n - 1}}$

Step 8

Simplify the numerator.

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Step 8.1

Factor $- 1$ out of ${(- 1)}^{n}$ .

$a_{n} = \frac{- {(- 1)}^{n - 1} \cdot 1^{n - 1}}{3 \cdot 2^{n - 1}}$

Step 8.2

Rewrite $- {(- 1)}^{n - 1}$ as $- 1 {(- 1)}^{n - 1}$ .

$a_{n} = \frac{- 1 {(- 1)}^{n - 1} \cdot 1^{n - 1}}{3 \cdot 2^{n - 1}}$

Step 8.3

Rewrite $1^{n - 1} {(- 1)}^{n - 1}$ as ${(1 \cdot - 1)}^{n - 1}$ .

$a_{n} = \frac{- 1 {(1 \cdot - 1)}^{n - 1}}{3 \cdot 2^{n - 1}}$

Step 9

Multiply $- 1$ by $1$ .

$a_{n} = \frac{- 1 {(- 1)}^{n - 1}}{3 \cdot 2^{n - 1}}$

Step 10

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

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Step 10.1

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

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Step 10.1.1

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

$a_{n} = \frac{{(- 1)}^{1} {(- 1)}^{n - 1}}{3 \cdot 2^{n - 1}}$

Step 10.1.2

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

$a_{n} = \frac{{(- 1)}^{1 + n - 1}}{3 \cdot 2^{n - 1}}$

Step 10.2

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

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Step 10.2.1

Subtract $1$ from $1$ .

$a_{n} = \frac{{(- 1)}^{n + 0}}{3 \cdot 2^{n - 1}}$

Step 10.2.2

Add $n$ and $0$ .

$a_{n} = \frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}$