Precalculus Examples

$4$ , $\frac{15}{4}$ , $\frac{7}{2}$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $- \frac{1}{4}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - \frac{1}{4}$

Step 2

This is the formula of an arithmetic sequence.

$a_{n} = a_{1} + d (n - 1)$

Step 3

Substitute in the values of $a_{1} = 4$ and $d = - \frac{1}{4}$ .

$a_{n} = 4 - \frac{1}{4} (n - 1)$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$a_{n} = 4 - \frac{1}{4} n - \frac{1}{4} \cdot - 1$

Step 4.2

Combine $n$ and $\frac{1}{4}$ .

$a_{n} = 4 - \frac{n}{4} - \frac{1}{4} \cdot - 1$

Step 4.3

Multiply $- \frac{1}{4} \cdot - 1$ .

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$a_{n} = 4 - \frac{n}{4} + 1 (\frac{1}{4})$

Step 4.3.2

Multiply $\frac{1}{4}$ by $1$ .

$a_{n} = 4 - \frac{n}{4} + \frac{1}{4}$

Step 5

To write $4$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$a_{n} = - \frac{n}{4} + 4 \cdot \frac{4}{4} + \frac{1}{4}$

Step 6

Combine $4$ and $\frac{4}{4}$ .

$a_{n} = - \frac{n}{4} + \frac{4 \cdot 4}{4} + \frac{1}{4}$

Step 7

Combine the numerators over the common denominator.

$a_{n} = - \frac{n}{4} + \frac{4 \cdot 4 + 1}{4}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $4$ by $4$ .

$a_{n} = - \frac{n}{4} + \frac{16 + 1}{4}$

Step 8.2

Add $16$ and $1$ .

$a_{n} = - \frac{n}{4} + \frac{17}{4}$