Calculus Examples

$18$ , $6$ , $2$

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

Geometric Sequence: $r = \frac{1}{3}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 18$ and $r = \frac{1}{3}$ .

$a_{n} = 18 {(\frac{1}{3})}^{n - 1}$

Step 4

Apply the product rule to $\frac{1}{3}$ .

$a_{n} = 18 \frac{1^{n - 1}}{3^{n - 1}}$

Step 5

One to any power is one.

$a_{n} = 18 \frac{1}{3^{n - 1}}$

Step 6

Combine $18$ and $\frac{1}{3^{n - 1}}$ .

$a_{n} = \frac{18}{3^{n - 1}}$

Step 7

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

$a_{4} = \frac{18}{3^{(4) - 1}}$

Step 8

Simplify the denominator.

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Step 8.1

Subtract $1$ from $4$ .

$a_{4} = \frac{18}{3^{3}}$

Step 8.2

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

$a_{4} = \frac{18}{27}$

Step 9

Cancel the common factor of $18$ and $27$ .

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Step 9.1

Factor $9$ out of $18$ .

$a_{4} = \frac{9 (2)}{27}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $9$ out of $27$ .

$a_{4} = \frac{9 \cdot 2}{9 \cdot 3}$

Step 9.2.2

Cancel the common factor.

$a_{4} = \frac{9 \cdot 2}{9 \cdot 3}$

Step 9.2.3

Rewrite the expression.

$a_{4} = \frac{2}{3}$

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