Algebra Examples

$\frac{1}{3}$ , $\frac{1}{6}$ , $\frac{1}{12}$

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

Geometric Sequence: $r = \frac{1}{2}$

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 $r = \frac{1}{2}$ .

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

Step 4

Apply the product rule to $\frac{1}{2}$ .

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

Step 5

Combine.

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

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

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

Step 6.1.2

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

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

Step 6.2

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

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Step 6.2.1

Subtract $1$ from $1$ .

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

Step 6.2.2

Add $n$ and $0$ .

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

Step 7

One to any power is one.

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