Algebra Examples

$\frac{2}{3}$ , $\frac{1}{3}$ , $\frac{1}{6}$ , $\frac{1}{12}$

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

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

Step 4

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

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

Step 5

Combine.

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

Step 6

One to any power is one.

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

Step 7

Multiply $2$ by $1$ .

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

Step 8

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

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

Step 9

Simplify the denominator.

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Step 9.1

Subtract $1$ from $5$ .

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

Step 9.2

Raise $2$ to the power of $4$ .

$a_{5} = \frac{2}{3 \cdot 16}$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Multiply $3$ by $16$ .

$a_{5} = \frac{2}{48}$

Step 10.2

Cancel the common factor of $2$ and $48$ .

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Step 10.2.1

Factor $2$ out of $2$ .

$a_{5} = \frac{2 (1)}{48}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $2$ out of $48$ .

$a_{5} = \frac{2 \cdot 1}{2 \cdot 24}$

Step 10.2.2.2

Cancel the common factor.

$a_{5} = \frac{2 \cdot 1}{2 \cdot 24}$

Step 10.2.2.3

Rewrite the expression.

$a_{5} = \frac{1}{24}$