Precalculus Examples

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

Step 1

Write $a_{n} = \frac{1}{3} {(3)}^{n - 1}$ as a function.

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

Step 2

Consider the difference quotient formula.

$\frac{f (n + h) - f n}{h}$

Step 3

Find the components of the definition.

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Step 3.1

Evaluate the function at $x = n + h$ .

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Step 3.1.1

Replace the variable $n$ with $n + h$ in the expression.

$f (n + h) = \frac{1}{3} \cdot {(3)}^{(n + h) - 1}$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

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

$f (n + h) = \frac{3^{n + h - 1}}{3}$

Step 3.1.2.2

Cancel the common factor of $3^{n + h - 1}$ and $3$ .

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Step 3.1.2.2.1

Factor $3$ out of $3^{n + h - 1}$ .

$f (n + h) = \frac{3 \cdot 3^{n + h - 2}}{3}$

Step 3.1.2.2.2

Cancel the common factors.

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Step 3.1.2.2.2.1

Factor $3$ out of $3$ .

$f (n + h) = \frac{3 \cdot 3^{n + h - 2}}{3 (1)}$

Step 3.1.2.2.2.2

Cancel the common factor.

$f (n + h) = \frac{3 \cdot 3^{n + h - 2}}{3 \cdot 1}$

Step 3.1.2.2.2.3

Rewrite the expression.

$f (n + h) = \frac{3^{n + h - 2}}{1}$

Step 3.1.2.2.2.4

Divide $3^{n + h - 2}$ by $1$ .

$f (n + h) = 3^{n + h - 2}$

Step 3.1.2.3

The final answer is $3^{n + h - 2}$ .

$3^{n + h - 2}$

Step 3.2

Find the components of the definition.

$f (n + h) = 3^{n + h - 2}$

$f (n) = 3^{n - 2}$

$f (n + h) = 3^{n + h - 2}$

$f (n) = 3^{n - 2}$

Step 4

Plug in the components.

$\frac{f (n + h) - f n}{h} = \frac{3^{n + h - 2} - (3^{n - 2})}{h}$

Step 5

Multiply $- 1$ by $3^{n - 2}$ .

$\frac{3^{n + h - 2} - 3^{n - 2}}{h}$

Step 6