Algebra Examples

$\frac{1}{3}$ , $\frac{3}{12}$ , $\frac{9}{48}$ , $\frac{27}{192}$

Step 1

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

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Step 1.1

Factor $3$ out of $3$ .

$\frac{1}{3}, \frac{3 (1)}{12}, \frac{9}{48}, \frac{27}{192}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $12$ .

$\frac{1}{3}, \frac{3 \cdot 1}{3 \cdot 4}, \frac{9}{48}, \frac{27}{192}$

Step 1.2.2

Cancel the common factor.

$\frac{1}{3}, \frac{3 \cdot 1}{3 \cdot 4}, \frac{9}{48}, \frac{27}{192}$

Step 1.2.3

Rewrite the expression.

$\frac{1}{3}, \frac{1}{4}, \frac{9}{48}, \frac{27}{192}$

Step 2

Cancel the common factor of $9$ and $48$ .

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Step 2.1

Factor $3$ out of $9$ .

$\frac{1}{3}, \frac{1}{4}, \frac{3 (3)}{48}, \frac{27}{192}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $3$ out of $48$ .

$\frac{1}{3}, \frac{1}{4}, \frac{3 \cdot 3}{3 \cdot 16}, \frac{27}{192}$

Step 2.2.2

Cancel the common factor.

$\frac{1}{3}, \frac{1}{4}, \frac{3 \cdot 3}{3 \cdot 16}, \frac{27}{192}$

Step 2.2.3

Rewrite the expression.

$\frac{1}{3}, \frac{1}{4}, \frac{3}{16}, \frac{27}{192}$

Step 3

Cancel the common factor of $27$ and $192$ .

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Step 3.1

Factor $3$ out of $27$ .

$\frac{1}{3}, \frac{1}{4}, \frac{3}{16}, \frac{3 (9)}{192}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $3$ out of $192$ .

$\frac{1}{3}, \frac{1}{4}, \frac{3}{16}, \frac{3 \cdot 9}{3 \cdot 64}$

Step 3.2.2

Cancel the common factor.

$\frac{1}{3}, \frac{1}{4}, \frac{3}{16}, \frac{3 \cdot 9}{3 \cdot 64}$

Step 3.2.3

Rewrite the expression.

$\frac{1}{3}, \frac{1}{4}, \frac{3}{16}, \frac{9}{64}$

Step 4

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 $\frac{3}{4}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{3}{4}$

Step 5

This is the form of a geometric sequence.

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

Step 6

Substitute in the values of $a_{1} = \frac{1}{3}$ and $r = \frac{3}{4}$ .

$a_{n} = \frac{1}{3} {(\frac{3}{4})}^{n - 1}$

Step 7

Apply the product rule to $\frac{3}{4}$ .

$a_{n} = \frac{1}{3} \cdot \frac{3^{n - 1}}{4^{n - 1}}$

Step 8

Combine.

$a_{n} = \frac{1 \cdot 3^{n - 1}}{3 \cdot 4^{n - 1}}$

Step 9

Cancel the common factor of $3^{n - 1}$ and $3$ .

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Step 9.1

Factor $3$ out of $1 \cdot 3^{n - 1}$ .

$a_{n} = \frac{3 (1 \cdot 3^{n - 2})}{3 \cdot 4^{n - 1}}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $3$ out of $3 \cdot 4^{n - 1}$ .

$a_{n} = \frac{3 (1 \cdot 3^{n - 2})}{3 (4^{n - 1})}$

Step 9.2.2

Cancel the common factor.

$a_{n} = \frac{3 (1 \cdot 3^{n - 2})}{3 \cdot 4^{n - 1}}$

Step 9.2.3

Rewrite the expression.

$a_{n} = \frac{1 \cdot 3^{n - 2}}{4^{n - 1}}$

Step 10

Multiply $3^{n - 2}$ by $1$ .

$a_{n} = \frac{3^{n - 2}}{4^{n - 1}}$

Step 11

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

$a_{5} = \frac{3^{(5) - 2}}{4^{(5) - 1}}$

Step 12

Simplify the numerator.

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Step 12.1

Subtract $2$ from $5$ .

$a_{5} = \frac{3^{3}}{4^{5 - 1}}$

Step 12.2

Raise $3$ to the power of $3$ .

$a_{5} = \frac{27}{4^{5 - 1}}$

Step 13

Simplify the denominator.

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Step 13.1

Subtract $1$ from $5$ .

$a_{5} = \frac{27}{4^{4}}$

Step 13.2

Raise $4$ to the power of $4$ .

$a_{5} = \frac{27}{256}$