Algebra Examples

$\frac{1}{2}$ , $\frac{3}{8}$ , $\frac{9}{32}$ , $\frac{27}{128}$

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 $\frac{3}{4}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{3}{4}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{1}{2}$ and $r = \frac{3}{4}$ .

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

Step 4

Apply the product rule to $\frac{3}{4}$ .

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

Step 5

Combine.

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

Step 6

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

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

Step 7

Simplify the denominator.

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Step 7.1

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

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

Step 7.2

Multiply the exponents in ${(2^{2})}^{n - 1}$ .

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Step 7.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.2.2

Apply the distributive property.

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

Step 7.2.3

Multiply $2$ by $- 1$ .

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

Step 7.3

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

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

Step 7.4

Subtract $2$ from $1$ .

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

Step 8

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

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

Step 9

Simplify the numerator.

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Step 9.1

Subtract $1$ from $5$ .

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

Step 9.2

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

$a_{5} = \frac{81}{2^{2 (5) - 1}}$

Step 10

Simplify the denominator.

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Step 10.1

Multiply $2$ by $5$ .

$a_{5} = \frac{81}{2^{10 - 1}}$

Step 10.2

Subtract $1$ from $10$ .

$a_{5} = \frac{81}{2^{9}}$

Step 10.3

Raise $2$ to the power of $9$ .

$a_{5} = \frac{81}{512}$