Precalculus Examples

\frac{1}{2}

$\frac{1}{2}$ ,

\frac{1}{6}

$\frac{1}{6}$ ,

\frac{1}{18}

$\frac{1}{18}$

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}{3}

$\frac{1}{3}$ gives the next term. In other words,

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

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

Geometric Sequence:

r = \frac{1}{3}

$r = \frac{1}{3}$

Step 2

This is the form of a geometric sequence.

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

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

Step 3

Substitute in the values of

a_{1} = \frac{1}{2}

$a_{1} = \frac{1}{2}$ and

r = \frac{1}{3}

$r = \frac{1}{3}$ .

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

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

Step 4

Apply the product rule to

\frac{1}{3}

$\frac{1}{3}$ .

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

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

Step 5

Combine.

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

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

Step 6

Multiply

1

$1$ by

1^{n - 1}

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

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Step 6.1

Multiply

1

$1$ by

1^{n - 1}

$1^{n - 1}$ .

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Step 6.1.1

Raise

1

$1$ to the power of

1

$1$ .

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

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

Step 6.1.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

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

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

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

Step 6.2

Combine the opposite terms in

1 + n - 1

$1 + n - 1$ .

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Step 6.2.1

Subtract

1

$1$ from

1

$1$ .

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

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

Step 6.2.2

Add

n

$n$ and

0

$0$ .

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

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

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

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

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

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

Step 7

One to any power is one.

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

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

Step 8

Substitute in the value of

n

$n$ to find the

n

$n$ th term.

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

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

Step 9

Simplify the denominator.

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Step 9.1

Subtract

1

$1$ from

4

$4$ .

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

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

Step 9.2

Raise

3

$3$ to the power of

3

$3$ .

a_{4} = \frac{1}{2 \cdot 27}

$a_{4} = \frac{1}{2 \cdot 27}$

a_{4} = \frac{1}{2 \cdot 27}

$a_{4} = \frac{1}{2 \cdot 27}$

Step 10

Multiply

2

$2$ by

27

$27$ .

a_{4} = \frac{1}{54}

$a_{4} = \frac{1}{54}$

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