Precalculus Examples

$\frac{2}{3}$ , $\frac{5}{6}$ , $1$ , $\frac{7}{6}$ , $\frac{4}{3}$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $\frac{1}{6}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = \frac{1}{6}$

Step 2

This is the formula of an arithmetic sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{2}{3}$ and $d = \frac{1}{6}$ .

$a_{n} = \frac{2}{3} + \frac{1}{6} (n - 1)$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$a_{n} = \frac{2}{3} + \frac{1}{6} n + \frac{1}{6} \cdot - 1$

Step 4.2

Combine $\frac{1}{6}$ and $n$ .

$a_{n} = \frac{2}{3} + \frac{n}{6} + \frac{1}{6} \cdot - 1$

Step 4.3

Combine $\frac{1}{6}$ and $- 1$ .

$a_{n} = \frac{2}{3} + \frac{n}{6} + \frac{- 1}{6}$

Step 4.4

Move the negative in front of the fraction.

$a_{n} = \frac{2}{3} + \frac{n}{6} - \frac{1}{6}$

Step 5

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$a_{n} = \frac{n}{6} + \frac{2}{3} \cdot \frac{2}{2} - \frac{1}{6}$

Step 6

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 6.1

Multiply $\frac{2}{3}$ by $\frac{2}{2}$ .

$a_{n} = \frac{n}{6} + \frac{2 \cdot 2}{3 \cdot 2} - \frac{1}{6}$

Step 6.2

Multiply $3$ by $2$ .

$a_{n} = \frac{n}{6} + \frac{2 \cdot 2}{6} - \frac{1}{6}$

Step 7

Combine the numerators over the common denominator.

$a_{n} = \frac{n}{6} + \frac{2 \cdot 2 - 1}{6}$

Step 8

Simplify the numerator.

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Step 8.1

Multiply $2$ by $2$ .

$a_{n} = \frac{n}{6} + \frac{4 - 1}{6}$

Step 8.2

Subtract $1$ from $4$ .

$a_{n} = \frac{n}{6} + \frac{3}{6}$

Step 9

Cancel the common factor of $3$ and $6$ .

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Step 9.1

Factor $3$ out of $3$ .

$a_{n} = \frac{n}{6} + \frac{3 (1)}{6}$

Step 9.2

Cancel the common factors.

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Step 9.2.1

Factor $3$ out of $6$ .

$a_{n} = \frac{n}{6} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 9.2.2

Cancel the common factor.

$a_{n} = \frac{n}{6} + \frac{3 \cdot 1}{3 \cdot 2}$

Step 9.2.3

Rewrite the expression.

$a_{n} = \frac{n}{6} + \frac{1}{2}$

Step 10

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

$a_{6} = \frac{6}{6} + \frac{1}{2}$

Step 11

Divide $6$ by $6$ .

$a_{6} = 1 + \frac{1}{2}$

Step 12

Write $1$ as a fraction with a common denominator.

$a_{6} = \frac{2}{2} + \frac{1}{2}$

Step 13

Combine the numerators over the common denominator.

$a_{6} = \frac{2 + 1}{2}$

Step 14

Add $2$ and $1$ .

$a_{6} = \frac{3}{2}$