Precalculus Examples

$2000$ , $200$ , $20$ , $2$

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 $\frac{1}{10}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{1}{10}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 2000$ and $r = \frac{1}{10}$ .

$a_{n} = 2000 {(\frac{1}{10})}^{n - 1}$

Step 4

Apply the product rule to $\frac{1}{10}$ .

$a_{n} = 2000 \frac{1^{n - 1}}{10^{n - 1}}$

Step 5

One to any power is one.

$a_{n} = 2000 \frac{1}{10^{n - 1}}$

Step 6

Combine $2000$ and $\frac{1}{10^{n - 1}}$ .

$a_{n} = \frac{2000}{10^{n - 1}}$

Step 7

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

$a_{6} = \frac{2000}{10^{(6) - 1}}$

Step 8

Simplify the denominator.

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Step 8.1

Subtract $1$ from $6$ .

$a_{6} = \frac{2000}{10^{5}}$

Step 8.2

Raise $10$ to the power of $5$ .

$a_{6} = \frac{2000}{100000}$

Step 9

Cancel the common factor of $2000$ and $100000$ .

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Step 9.1

Factor $2000$ out of $2000$ .

$a_{6} = \frac{2000 (1)}{100000}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $2000$ out of $100000$ .

$a_{6} = \frac{2000 \cdot 1}{2000 \cdot 50}$

Step 9.2.2

Cancel the common factor.

$a_{6} = \frac{2000 \cdot 1}{2000 \cdot 50}$

Step 9.2.3

Rewrite the expression.

$a_{6} = \frac{1}{50}$