Algebra Examples

$\frac{13}{15}$ , $\frac{11}{15}$ , $\frac{3}{5}$ , $\frac{7}{15}$

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 $a_{1} = \frac{13}{15}$ and $d = - \frac{2}{15}$ .

$a_{n} = \frac{13}{15} - \frac{2}{15} (n - 1)$

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

$a_{n} = \frac{13}{15} - \frac{2}{15} n - \frac{2}{15} \cdot - 1$

Step 4.2

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

$a_{n} = \frac{13}{15} - \frac{n \cdot 2}{15} - \frac{2}{15} \cdot - 1$

Step 4.3

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

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

$a_{n} = \frac{13}{15} - \frac{n \cdot 2}{15} + 1 (\frac{2}{15})$

Step 4.3.2

Multiply $\frac{2}{15}$ by $1$ .

$a_{n} = \frac{13}{15} - \frac{n \cdot 2}{15} + \frac{2}{15}$

Step 4.4

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

$a_{n} = \frac{13}{15} - \frac{2 n}{15} + \frac{2}{15}$

Step 5

Combine the numerators over the common denominator.

$a_{n} = \frac{13 - 2 n + 2}{15}$

Step 6

Add $13$ and $2$ .

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

Step 7

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

$a_{5} = \frac{- 2 \cdot 5 + 15}{15}$

Step 8

Cancel the common factor of $- 2 \cdot 5 + 15$ and $15$ .

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Step 8.1

Factor $5$ out of $- 2 \cdot 5$ .

$a_{5} = \frac{5 \cdot - 2 + 15}{15}$

Step 8.2

Factor $5$ out of $15$ .

$a_{5} = \frac{5 \cdot - 2 + 5 \cdot 3}{15}$

Step 8.3

Factor $5$ out of $5 \cdot - 2 + 5 \cdot 3$ .

$a_{5} = \frac{5 \cdot (- 2 + 3)}{15}$

Step 8.4

Cancel the common factors.

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Step 8.4.1

Factor $5$ out of $15$ .

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

Step 8.4.2

Cancel the common factor.

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

Step 8.4.3

Rewrite the expression.

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

Step 9

Add $- 2$ and $3$ .

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