Precalculus Examples

$a_{n} = 12 + (n - 1) (- 8)$

Step 1

Write $a_{n} = 12 + (n - 1) (- 8)$ as a function.

$f (n) = 12 + (n - 1) (- 8)$

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.

$f (n + h) = 12 + ((n + h) - 1) (- 8)$

Step 3.1.2

Simplify the result.

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Step 3.1.2.1

Simplify each term.

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Step 3.1.2.1.1

Apply the distributive property.

$f (n + h) = 12 + n \cdot - 8 + h \cdot - 8 - 1 \cdot - 8$

Step 3.1.2.1.2

Simplify.

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Step 3.1.2.1.2.1

Move $- 8$ to the left of $n$ .

$f (n + h) = 12 - 8 \cdot n + h \cdot - 8 - 1 \cdot - 8$

Step 3.1.2.1.2.2

Move $- 8$ to the left of $h$ .

$f (n + h) = 12 - 8 \cdot n - 8 \cdot h - 1 \cdot - 8$

Step 3.1.2.1.2.3

Multiply $- 1$ by $- 8$ .

$f (n + h) = 12 - 8 \cdot n - 8 \cdot h + 8$

Step 3.1.2.1.3

Multiply $- 8$ by $h$ .

$f (n + h) = 12 - 8 n - 8 h + 8$

Step 3.1.2.2

Add $12$ and $8$ .

$f (n + h) = - 8 n - 8 h + 20$

Step 3.1.2.3

The final answer is $- 8 n - 8 h + 20$ .

$- 8 n - 8 h + 20$

Step 3.2

Reorder $- 8 n$ and $- 8 h$ .

$- 8 h - 8 n + 20$

Step 3.3

Find the components of the definition.

$f (n + h) = - 8 h - 8 n + 20$

$f (n) = - 8 n + 20$

$f (n + h) = - 8 h - 8 n + 20$

$f (n) = - 8 n + 20$

Step 4

Plug in the components.

$\frac{f (n + h) - f n}{h} = \frac{- 8 h - 8 n + 20 - (- 8 n + 20)}{h}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Factor $4$ out of $- 8 n + 20$ .

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Step 5.1.1.1

Factor $4$ out of $- 8 n$ .

$\frac{- 8 h - 8 n + 20 - (4 (- 2 n) + 20)}{h}$

Step 5.1.1.2

Factor $4$ out of $20$ .

$\frac{- 8 h - 8 n + 20 - (4 (- 2 n) + 4 (5))}{h}$

Step 5.1.1.3

Factor $4$ out of $4 (- 2 n) + 4 (5)$ .

$\frac{- 8 h - 8 n + 20 - (4 (- 2 n + 5))}{h}$

$\frac{- 8 h - 8 n + 20 - 1 \cdot 4 (- 2 n + 5)}{h}$

Step 5.1.2

Multiply $- 1$ by $4$ .

$\frac{- 8 h - 8 n + 20 - 4 (- 2 n + 5)}{h}$

Step 5.1.3

Factor $4$ out of $- 8 h - 8 n + 20 - 4 (- 2 n + 5)$ .

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Step 5.1.3.1

Factor $4$ out of $- 8 h$ .

$\frac{4 (- 2 h) - 8 n + 20 - 4 (- 2 n + 5)}{h}$

Step 5.1.3.2

Factor $4$ out of $- 8 n$ .

$\frac{4 (- 2 h) + 4 (- 2 n) + 20 - 4 (- 2 n + 5)}{h}$

Step 5.1.3.3

Factor $4$ out of $20$ .

$\frac{4 (- 2 h) + 4 (- 2 n) + 4 (5) - 4 (- 2 n + 5)}{h}$

Step 5.1.3.4

Factor $4$ out of $- 4 (- 2 n + 5)$ .

$\frac{4 (- 2 h) + 4 (- 2 n) + 4 (5) + 4 (- (- 2 n + 5))}{h}$

Step 5.1.3.5

Factor $4$ out of $4 (- 2 h) + 4 (- 2 n)$ .

$\frac{4 (- 2 h - 2 n) + 4 (5) + 4 (- (- 2 n + 5))}{h}$

Step 5.1.3.6

Factor $4$ out of $4 (- 2 h - 2 n) + 4 (5)$ .

$\frac{4 (- 2 h - 2 n + 5) + 4 (- (- 2 n + 5))}{h}$

Step 5.1.3.7

Factor $4$ out of $4 (- 2 h - 2 n + 5) + 4 (- (- 2 n + 5))$ .

$\frac{4 (- 2 h - 2 n + 5 - (- 2 n + 5))}{h}$

Step 5.1.4

Apply the distributive property.

$\frac{4 (- 2 h - 2 n + 5 - (- 2 n) - 1 \cdot 5)}{h}$

Step 5.1.5

Multiply $- 2$ by $- 1$ .

$\frac{4 (- 2 h - 2 n + 5 + 2 n - 1 \cdot 5)}{h}$

Step 5.1.6

Multiply $- 1$ by $5$ .

$\frac{4 (- 2 h - 2 n + 5 + 2 n - 5)}{h}$

Step 5.1.7

Add $- 2 n$ and $2 n$ .

$\frac{4 (- 2 h + 0 + 5 - 5)}{h}$

Step 5.1.8

Add $- 2 h$ and $0$ .

$\frac{4 (- 2 h + 5 - 5)}{h}$

Step 5.1.9

Subtract $5$ from $5$ .

$\frac{4 (- 2 h + 0)}{h}$

Step 5.1.10

Add $- 2 h$ and $0$ .

$\frac{4 \cdot - 2 h}{h}$

Step 5.1.11

Multiply $4$ by $- 2$ .

$\frac{- 8 h}{h}$

Step 5.2

Cancel the common factor of $h$ .

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Step 5.2.1

Cancel the common factor.

$\frac{- 8 h}{h}$

Step 5.2.2

Divide $- 8$ by $1$ .

$- 8$

Step 6