Algebra Examples

$\frac{4}{5}$ , $\frac{8}{15}$ , $\frac{16}{45}$ , $\frac{32}{135}$

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 $a_{1} = \frac{4}{5}$ and $r = \frac{2}{3}$ .

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

Step 4

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

$a_{n} = \frac{4}{5} \cdot \frac{2^{n - 1}}{3^{n - 1}}$

Step 5

Combine.

$a_{n} = \frac{4 \cdot 2^{n - 1}}{5 \cdot 3^{n - 1}}$

Step 6

Simplify the numerator.

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Step 6.1

Rewrite $4$ as $2^{2}$ .

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

Step 6.2

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

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

Step 6.3

Subtract $1$ from $2$ .

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

Step 7

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

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

Step 8

Simplify the numerator.

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Step 8.1

Add $5$ and $1$ .

$a_{5} = \frac{2^{6}}{5 \cdot 3^{5 - 1}}$

Step 8.2

Raise $2$ to the power of $6$ .

$a_{5} = \frac{64}{5 \cdot 3^{5 - 1}}$

Step 9

Simplify the denominator.

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Step 9.1

Subtract $1$ from $5$ .

$a_{5} = \frac{64}{5 \cdot 3^{4}}$

Step 9.2

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

$a_{5} = \frac{64}{5 \cdot 81}$

Step 10

Multiply $5$ by $81$ .

$a_{5} = \frac{64}{405}$