Algebra Examples

$\frac{1}{2}$ , $\frac{7}{6}$ , $\frac{11}{6}$ , $\frac{5}{2}$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $\frac{2}{3}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = \frac{2}{3}$

Step 2

This is the formula of an arithmetic sequence.

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

Step 3

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

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

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine $\frac{2}{3}$ and $n$ .

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

Step 4.3

Multiply $\frac{2}{3} \cdot - 1$ .

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Step 4.3.1

Combine $\frac{2}{3}$ and $- 1$ .

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

Step 4.3.2

Multiply $2$ by $- 1$ .

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

Step 4.4

Move the negative in front of the fraction.

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

Step 5

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 6

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

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

Step 7

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

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Step 7.1

Multiply $\frac{1}{2}$ by $\frac{3}{3}$ .

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

Step 7.2

Multiply $2$ by $3$ .

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

Step 7.3

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

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

Step 7.4

Multiply $3$ by $2$ .

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

Step 8

Combine the numerators over the common denominator.

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

Step 9

Simplify the numerator.

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Step 9.1

Multiply $- 2$ by $2$ .

$a_{n} = \frac{2 n}{3} + \frac{3 - 4}{6}$

Step 9.2

Subtract $4$ from $3$ .

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

Step 10

Move the negative in front of the fraction.

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