Find the 6th Term 27/375 , 9/75 , 3/15 , 1/3

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

Step 1

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.

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

Step 2

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.

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

Step 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.

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

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

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

Step 5

This is the form of a geometric sequence.

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

Step 6

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

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

Step 7

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

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

Step 8

Combine.

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

Step 9

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

$a_{6} = \frac{9 \cdot 5^{(6) - 1}}{125 \cdot 3^{(6) - 1}}$

Step 10

Simplify the numerator.

Subtract $1$ from $6$ .

$a_{6} = \frac{9 \cdot 5^{5}}{125 \cdot 3^{6 - 1}}$

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

$a_{6} = \frac{9 \cdot 3125}{125 \cdot 3^{6 - 1}}$

Step 11

Simplify the denominator.

Subtract $1$ from $6$ .

$a_{6} = \frac{9 \cdot 3125}{125 \cdot 3^{5}}$

Raise $3$ to the power of $5$ .

$a_{6} = \frac{9 \cdot 3125}{125 \cdot 243}$

Step 12

Reduce the expression by cancelling the common factors.

Multiply $9$ by $3125$ .

$a_{6} = \frac{28125}{125 \cdot 243}$

Multiply $125$ by $243$ .

$a_{6} = \frac{28125}{30375}$

Cancel the common factor of $28125$ and $30375$ .

Factor $1125$ out of $28125$ .

$a_{6} = \frac{1125 (25)}{30375}$

Cancel the common factors.

Factor $1125$ out of $30375$ .

$a_{6} = \frac{1125 \cdot 25}{1125 \cdot 27}$

Cancel the common factor.

$a_{6} = \frac{1125 \cdot 25}{1125 \cdot 27}$

Rewrite the expression.

$a_{6} = \frac{25}{27}$

Find the 6th Term 27/375 , 9/75 , 3/15 , 1/3

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