Identify the Progression 1/3 , 2/9 , 4/27 , 8/81 , 16/243

$\frac{1}{3}$ , $\frac{2}{9}$ , $\frac{4}{27}$ , $\frac{8}{81}$ , $\frac{16}{243}$

Step 1

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 $\frac{2}{3}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{2}{3}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{1}{3}$ and $r = \frac{2}{3}$ .

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

Step 4

Apply the product rule to $\frac{2}{3}$ .

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

Step 5

Combine.

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

Step 6

Multiply $3$ by $3^{n - 1}$ by adding the exponents.

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

Raise $3$ to the power of $1$ .

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

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Combine the opposite terms in $1 + n - 1$ .

Subtract $1$ from $1$ .

$a_{n} = \frac{1 \cdot 2^{n - 1}}{3^{n + 0}}$

Add $n$ and $0$ .

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

Step 7

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

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

Identify the Progression 1/3 , 2/9 , 4/27 , 8/81 , 16/243

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