Find the 6th Term 2/3 , 4/3 , 2 , 8/3

$\frac{2}{3}$ , $\frac{4}{3}$ , $2$ , $\frac{8}{3}$

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 $a_{1} = \frac{2}{3}$ and $d = \frac{2}{3}$ .

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

Step 4

Simplify each term.

Apply the distributive property.

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

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

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

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

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

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

Multiply $2$ by $- 1$ .

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

Move the negative in front of the fraction.

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

Step 5

Combine the opposite terms in $\frac{2}{3} + \frac{2 n}{3} - \frac{2}{3}$ .

Combine the numerators over the common denominator.

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

Subtract $2$ from $2$ .

$a_{n} = \frac{2 n}{3} + \frac{0}{3}$

Divide $0$ by $3$ .

$a_{n} = \frac{2 n}{3} + 0$

Add $\frac{2 n}{3}$ and $0$ .

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

Step 6

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

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

Step 7

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

Factor $3$ out of $2 (6)$ .

$a_{6} = \frac{3 (2 \cdot (2))}{3}$

Cancel the common factors.

Factor $3$ out of $3$ .

$a_{6} = \frac{3 (2 \cdot (2))}{3 (1)}$

Cancel the common factor.

$a_{6} = \frac{3 (2 \cdot (2))}{3 \cdot 1}$

Rewrite the expression.

$a_{6} = \frac{2 \cdot (2)}{1}$

Divide $2 \cdot (2)$ by $1$ .

$a_{6} = 2 \cdot (2)$

Step 8

Multiply $2$ by $2$ .

$a_{6} = 4$

Find the 6th Term 2/3 , 4/3 , 2 , 8/3

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