Find the Next Term 27/375 , 9/75 , 3/15 , 1/3

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

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

Factor $3$ out of $27$ .

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

Cancel the common factors.

Factor $3$ out of $375$ .

$\frac{3 \cdot 9}{3 \cdot 125}, \frac{9}{75}, \frac{3}{15}, \frac{1}{3}$

Cancel the common factor.

$\frac{3 \cdot 9}{3 \cdot 125}, \frac{9}{75}, \frac{3}{15}, \frac{1}{3}$

Rewrite the expression.

$\frac{9}{125}, \frac{9}{75}, \frac{3}{15}, \frac{1}{3}$

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

Factor $3$ out of $9$ .

$\frac{9}{125}, \frac{3 (3)}{75}, \frac{3}{15}, \frac{1}{3}$

Cancel the common factors.

Factor $3$ out of $75$ .

$\frac{9}{125}, \frac{3 \cdot 3}{3 \cdot 25}, \frac{3}{15}, \frac{1}{3}$

Cancel the common factor.

$\frac{9}{125}, \frac{3 \cdot 3}{3 \cdot 25}, \frac{3}{15}, \frac{1}{3}$

Rewrite the expression.

$\frac{9}{125}, \frac{3}{25}, \frac{3}{15}, \frac{1}{3}$

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

Factor $3$ out of $3$ .

$\frac{9}{125}, \frac{3}{25}, \frac{3 (1)}{15}, \frac{1}{3}$

Cancel the common factors.

Factor $3$ out of $15$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

This is the form of a geometric sequence.

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

Substitute in the values of $a_{1} = \frac{9}{125}$ and $r = \frac{5}{3}$ .

$a_{n} = \frac{9}{125} {(\frac{5}{3})}^{n - 1}$

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

$a_{n} = \frac{9}{125} \cdot \frac{5^{n - 1}}{3^{n - 1}}$

Combine.

$a_{n} = \frac{9 \cdot 5^{n - 1}}{125 \cdot 3^{n - 1}}$

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

$a_{5} = \frac{9 \cdot 5^{(5) - 1}}{125 \cdot 3^{(5) - 1}}$

Simplify the numerator.

Subtract $1$ from $5$ .

$a_{5} = \frac{9 \cdot 5^{4}}{125 \cdot 3^{5 - 1}}$

Raise $5$ to the power of $4$ .

$a_{5} = \frac{9 \cdot 625}{125 \cdot 3^{5 - 1}}$

Simplify the denominator.

Subtract $1$ from $5$ .

$a_{5} = \frac{9 \cdot 625}{125 \cdot 3^{4}}$

Raise $3$ to the power of $4$ .

$a_{5} = \frac{9 \cdot 625}{125 \cdot 81}$

Reduce the expression by cancelling the common factors.

Multiply $9$ by $625$ .

$a_{5} = \frac{5625}{125 \cdot 81}$

Multiply $125$ by $81$ .

$a_{5} = \frac{5625}{10125}$

Cancel the common factor of $5625$ and $10125$ .

Factor $1125$ out of $5625$ .

$a_{5} = \frac{1125 (5)}{10125}$

Cancel the common factors.

Factor $1125$ out of $10125$ .

$a_{5} = \frac{1125 \cdot 5}{1125 \cdot 9}$

Cancel the common factor.

$a_{5} = \frac{1125 \cdot 5}{1125 \cdot 9}$

Rewrite the expression.

$a_{5} = \frac{5}{9}$

Find the Next Term 27/375 , 9/75 , 3/15 , 1/3

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