Algebra Examples

$5$ , $\frac{13}{3}$ , $\frac{11}{3}$

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: $d = - \frac{2}{3}$

This is the formula of an arithmetic sequence.

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

Substitute in the values of $a_{1} = 5$ and $d = - \frac{2}{3}$ .

$a_{n} = 5 - \frac{2}{3} (n - 1)$

Simplify each term.

Tap for more steps...

Apply the distributive property.

$a_{n} = 5 - \frac{2}{3} n - \frac{2}{3} \cdot - 1$

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

$a_{n} = 5 - \frac{n \cdot 2}{3} - \frac{2}{3} \cdot - 1$

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

Tap for more steps...

Multiply $- 1$ by $- 1$ .

$a_{n} = 5 - \frac{n \cdot 2}{3} + 1 (\frac{2}{3})$

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

$a_{n} = 5 - \frac{n \cdot 2}{3} + \frac{2}{3}$

Move $2$ to the left of $n$ .

$a_{n} = 5 - \frac{2 n}{3} + \frac{2}{3}$

To write $5$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a_{n} = - \frac{2 n}{3} + 5 \cdot \frac{3}{3} + \frac{2}{3}$

Combine $5$ and $\frac{3}{3}$ .

$a_{n} = - \frac{2 n}{3} + \frac{5 \cdot 3}{3} + \frac{2}{3}$

Combine the numerators over the common denominator.

$a_{n} = - \frac{2 n}{3} + \frac{5 \cdot 3 + 2}{3}$

Simplify the numerator.

Tap for more steps...

Multiply $5$ by $3$ .

$a_{n} = - \frac{2 n}{3} + \frac{15 + 2}{3}$

Add $15$ and $2$ .

$a_{n} = - \frac{2 n}{3} + \frac{17}{3}$

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

$a_{4} = - \frac{2 (4)}{3} + \frac{17}{3}$

Combine the numerators over the common denominator.

$a_{4} = \frac{- 2 \cdot 4 + 17}{3}$

Simplify the expression.

Tap for more steps...

Multiply $- 2$ by $4$ .

$a_{4} = \frac{- 8 + 17}{3}$

Add $- 8$ and $17$ .

$a_{4} = \frac{9}{3}$

Divide $9$ by $3$ .

$a_{4} = 3$