Algebra Examples

$\frac{4}{15}$ , $\frac{2}{5}$ , $\frac{8}{15}$ , $\frac{2}{3}$

Step 1

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $\frac{2}{15}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = \frac{2}{15}$

Step 2

This is the formula of an arithmetic sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{4}{15}$ and $d = \frac{2}{15}$ .

$a_{n} = \frac{4}{15} + \frac{2}{15} (n - 1)$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$a_{n} = \frac{4}{15} + \frac{2}{15} n + \frac{2}{15} \cdot - 1$

Step 4.2

Combine $\frac{2}{15}$ and $n$ .

$a_{n} = \frac{4}{15} + \frac{2 n}{15} + \frac{2}{15} \cdot - 1$

Step 4.3

Multiply $\frac{2}{15} \cdot - 1$ .

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Step 4.3.1

Combine $\frac{2}{15}$ and $- 1$ .

$a_{n} = \frac{4}{15} + \frac{2 n}{15} + \frac{2 \cdot - 1}{15}$

Step 4.3.2

Multiply $2$ by $- 1$ .

$a_{n} = \frac{4}{15} + \frac{2 n}{15} + \frac{- 2}{15}$

Step 4.4

Move the negative in front of the fraction.

$a_{n} = \frac{4}{15} + \frac{2 n}{15} - \frac{2}{15}$

Step 5

Combine the numerators over the common denominator.

$a_{n} = \frac{4 + 2 n - 2}{15}$

Step 6

Subtract $2$ from $4$ .

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

Step 7

Factor $2$ out of $2 n + 2$ .

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Step 7.1

Factor $2$ out of $2 n$ .

$a_{n} = \frac{2 (n) + 2}{15}$

Step 7.2

Factor $2$ out of $2$ .

$a_{n} = \frac{2 (n) + 2 (1)}{15}$

Step 7.3

Factor $2$ out of $2 (n) + 2 (1)$ .

$a_{n} = \frac{2 (n + 1)}{15}$

Step 8

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

$a_{5} = \frac{2 ((5) + 1)}{15}$

Step 9

Simplify the expression.

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Step 9.1

Add $5$ and $1$ .

$a_{5} = \frac{2 \cdot 6}{15}$

Step 9.2

Multiply $2$ by $6$ .

$a_{5} = \frac{12}{15}$

Step 10

Cancel the common factor of $12$ and $15$ .

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Step 10.1

Factor $3$ out of $12$ .

$a_{5} = \frac{3 (4)}{15}$

Step 10.2

Cancel the common factors.

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Step 10.2.1

Factor $3$ out of $15$ .

$a_{5} = \frac{3 \cdot 4}{3 \cdot 5}$

Step 10.2.2

Cancel the common factor.

$a_{5} = \frac{3 \cdot 4}{3 \cdot 5}$

Step 10.2.3

Rewrite the expression.

$a_{5} = \frac{4}{5}$