Find the 5th Term 3 , 7/3 , 5/3

$3$ , $\frac{7}{3}$ , $\frac{5}{3}$

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

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

Step 4

Simplify each term.

Apply the distributive property.

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

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

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

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

Multiply $- 1$ by $- 1$ .

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

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

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

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

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

Step 5

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

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

Step 6

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

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

Step 7

Combine the numerators over the common denominator.

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

Step 8

Simplify the numerator.

Multiply $3$ by $3$ .

$a_{n} = - \frac{2 n}{3} + \frac{9 + 2}{3}$

Add $9$ and $2$ .

$a_{n} = - \frac{2 n}{3} + \frac{11}{3}$

Step 9

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

$a_{5} = - \frac{2 (5)}{3} + \frac{11}{3}$

Step 10

Combine the numerators over the common denominator.

$a_{5} = \frac{- 2 \cdot 5 + 11}{3}$

Step 11

Simplify the expression.

Multiply $- 2$ by $5$ .

$a_{5} = \frac{- 10 + 11}{3}$

Add $- 10$ and $11$ .

$a_{5} = \frac{1}{3}$

Find the 5th Term 3 , 7/3 , 5/3

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