Precalculus Examples

$\frac{3}{3}$ , $\frac{3}{12}$ , $\frac{3}{48}$ , $\frac{3}{192}$ , $\frac{3}{768}$

Step 1

Divide $3$ by $3$ .

$1, \frac{3}{12}, \frac{3}{48}, \frac{3}{192}, \frac{3}{768}$

Step 2

Cancel the common factor of $3$ and $12$ .

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Step 2.1

Factor $3$ out of $3$ .

$1, \frac{3 (1)}{12}, \frac{3}{48}, \frac{3}{192}, \frac{3}{768}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $12$ .

$1, \frac{3 \cdot 1}{3 \cdot 4}, \frac{3}{48}, \frac{3}{192}, \frac{3}{768}$

Step 2.2.2

Cancel the common factor.

$1, \frac{3 \cdot 1}{3 \cdot 4}, \frac{3}{48}, \frac{3}{192}, \frac{3}{768}$

Step 2.2.3

Rewrite the expression.

$1, \frac{1}{4}, \frac{3}{48}, \frac{3}{192}, \frac{3}{768}$

Step 3

Cancel the common factor of $3$ and $48$ .

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Step 3.1

Factor $3$ out of $3$ .

$1, \frac{1}{4}, \frac{3 (1)}{48}, \frac{3}{192}, \frac{3}{768}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $48$ .

$1, \frac{1}{4}, \frac{3 \cdot 1}{3 \cdot 16}, \frac{3}{192}, \frac{3}{768}$

Step 3.2.2

Cancel the common factor.

$1, \frac{1}{4}, \frac{3 \cdot 1}{3 \cdot 16}, \frac{3}{192}, \frac{3}{768}$

Step 3.2.3

Rewrite the expression.

$1, \frac{1}{4}, \frac{1}{16}, \frac{3}{192}, \frac{3}{768}$

Step 4

Cancel the common factor of $3$ and $192$ .

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Step 4.1

Factor $3$ out of $3$ .

$1, \frac{1}{4}, \frac{1}{16}, \frac{3 (1)}{192}, \frac{3}{768}$

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $192$ .

$1, \frac{1}{4}, \frac{1}{16}, \frac{3 \cdot 1}{3 \cdot 64}, \frac{3}{768}$

Step 4.2.2

Cancel the common factor.

$1, \frac{1}{4}, \frac{1}{16}, \frac{3 \cdot 1}{3 \cdot 64}, \frac{3}{768}$

Step 4.2.3

Rewrite the expression.

$1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{3}{768}$

Step 5

Cancel the common factor of $3$ and $768$ .

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Step 5.1

Factor $3$ out of $3$ .

$1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{3 (1)}{768}$

Step 5.2

Cancel the common factors.

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Step 5.2.1

Factor $3$ out of $768$ .

$1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{3 \cdot 1}{3 \cdot 256}$

Step 5.2.2

Cancel the common factor.

$1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{3 \cdot 1}{3 \cdot 256}$

Step 5.2.3

Rewrite the expression.

$1, \frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}$

Step 6

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 $\frac{1}{4}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{1}{4}$

Step 7

This is the form of a geometric sequence.

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

Step 8

Substitute in the values of $a_{1} = 1$ and $r = \frac{1}{4}$ .

$a_{n} = 1 {(\frac{1}{4})}^{n - 1}$

Step 9

Multiply ${(\frac{1}{4})}^{n - 1}$ by $1$ .

$a_{n} = {(\frac{1}{4})}^{n - 1}$

Step 10

Apply the product rule to $\frac{1}{4}$ .

$a_{n} = \frac{1^{n - 1}}{4^{n - 1}}$

Step 11

One to any power is one.

$a_{n} = \frac{1}{4^{n - 1}}$