Identify the Sequence 1/3 , 3/15 , 9/75 , 27/375

$\frac{1}{3}$ , $\frac{3}{15}$ , $\frac{9}{75}$ , $\frac{27}{375}$

Step 1

Cancel the common factor of $3$ and $15$ .

Factor $3$ out of $3$ .

$\frac{1}{3}, \frac{3 (1)}{15}, \frac{9}{75}, \frac{27}{375}$

Cancel the common factors.

Factor $3$ out of $15$ .

$\frac{1}{3}, \frac{3 \cdot 1}{3 \cdot 5}, \frac{9}{75}, \frac{27}{375}$

Cancel the common factor.

$\frac{1}{3}, \frac{3 \cdot 1}{3 \cdot 5}, \frac{9}{75}, \frac{27}{375}$

Rewrite the expression.

$\frac{1}{3}, \frac{1}{5}, \frac{9}{75}, \frac{27}{375}$

Step 2

Cancel the common factor of $9$ and $75$ .

Factor $3$ out of $9$ .

$\frac{1}{3}, \frac{1}{5}, \frac{3 (3)}{75}, \frac{27}{375}$

Cancel the common factors.

Factor $3$ out of $75$ .

$\frac{1}{3}, \frac{1}{5}, \frac{3 \cdot 3}{3 \cdot 25}, \frac{27}{375}$

Cancel the common factor.

$\frac{1}{3}, \frac{1}{5}, \frac{3 \cdot 3}{3 \cdot 25}, \frac{27}{375}$

Rewrite the expression.

$\frac{1}{3}, \frac{1}{5}, \frac{3}{25}, \frac{27}{375}$

Step 3

Cancel the common factor of $27$ and $375$ .

Factor $3$ out of $27$ .

$\frac{1}{3}, \frac{1}{5}, \frac{3}{25}, \frac{3 (9)}{375}$

Cancel the common factors.

Factor $3$ out of $375$ .

$\frac{1}{3}, \frac{1}{5}, \frac{3}{25}, \frac{3 \cdot 9}{3 \cdot 125}$

Cancel the common factor.

$\frac{1}{3}, \frac{1}{5}, \frac{3}{25}, \frac{3 \cdot 9}{3 \cdot 125}$

Rewrite the expression.

$\frac{1}{3}, \frac{1}{5}, \frac{3}{25}, \frac{9}{125}$

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 $\frac{3}{5}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{3}{5}$

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 $r = \frac{3}{5}$ .

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

Step 7

Apply the product rule to $\frac{3}{5}$ .

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

Step 8

Combine.

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

Step 9

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

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

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

Cancel the common factors.

Factor $3$ out of $3 \cdot 5^{n - 1}$ .

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

Cancel the common factor.

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

Rewrite the expression.

$a_{n} = \frac{1 \cdot 3^{n - 2}}{5^{n - 1}}$

Step 10

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

$a_{n} = \frac{3^{n - 2}}{5^{n - 1}}$

Identify the Sequence 1/3 , 3/15 , 9/75 , 27/375

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