Precalculus Examples

$\frac{1}{2}$ , $- \frac{1}{10}$ , $\frac{1}{50}$ , $- \frac{1}{250}$

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 $- \frac{1}{5}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = - \frac{1}{5}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{1}{2}$ and $r = - \frac{1}{5}$ .

$a_{n} = \frac{1}{2} {(- \frac{1}{5})}^{n - 1}$

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 $- \frac{1}{5}$ .

$a_{n} = \frac{1}{2} ({(- 1)}^{n - 1} {(\frac{1}{5})}^{n - 1})$

Step 4.2

Apply the product rule to $\frac{1}{5}$ .

$a_{n} = \frac{1}{2} ({(- 1)}^{n - 1} \frac{1^{n - 1}}{5^{n - 1}})$

Step 5

One to any power is one.

$a_{n} = \frac{1}{2} ({(- 1)}^{n - 1} \frac{1}{5^{n - 1}})$

Step 6

Combine ${(- 1)}^{n - 1}$ and $\frac{1}{5^{n - 1}}$ .

$a_{n} = \frac{1}{2} \cdot \frac{{(- 1)}^{n - 1}}{5^{n - 1}}$

Step 7

Combine.

$a_{n} = \frac{1 {(- 1)}^{n - 1}}{2 \cdot 5^{n - 1}}$

Step 8

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

$a_{n} = \frac{{(- 1)}^{n - 1}}{2 \cdot 5^{n - 1}}$