Precalculus Examples

$\frac{1}{2}$ , $1$ , $2$ , $4$ , $8$

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

Geometric Sequence: $r = 2$

Step 2

This is the form of a geometric sequence.

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

Step 3

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

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

Step 4

Combine $\frac{1}{2}$ and $2^{n - 1}$ .

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

Step 5

Cancel the common factor of $2^{n - 1}$ and $2$ .

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Step 5.1

Factor $2$ out of $2^{n - 1}$ .

$a_{n} = \frac{2 \cdot 2^{n - 2}}{2}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $2$ out of $2$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.2.4

Divide $2^{n - 2}$ by $1$ .

$a_{n} = 2^{n - 2}$