Algebra Examples

$\frac{1}{3}$ , $\frac{2}{9}$ , $\frac{3}{27}$

Step 1

Cancel the common factor of $3$ and $27$ .

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Step 1.1

Factor $3$ out of $3$ .

$\frac{1}{3}, \frac{2}{9}, \frac{3 (1)}{27}$

Step 1.2

Cancel the common factors.

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Step 1.2.1

Factor $3$ out of $27$ .

$\frac{1}{3}, \frac{2}{9}, \frac{3 \cdot 1}{3 \cdot 9}$

Step 1.2.2

Cancel the common factor.

$\frac{1}{3}, \frac{2}{9}, \frac{3 \cdot 1}{3 \cdot 9}$

Step 1.2.3

Rewrite the expression.

$\frac{1}{3}, \frac{2}{9}, \frac{1}{9}$

Step 2

This is an arithmetic sequence since there is a common difference between each term. In this case, adding $- \frac{1}{9}$ to the previous term in the sequence gives the next term. In other words, $a_{n} = a_{1} + d (n - 1)$ .

Arithmetic Sequence: $d = - \frac{1}{9}$

Step 3

This is the formula of an arithmetic sequence.

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

Step 4

Substitute in the values of $a_{1} = \frac{1}{3}$ and $d = - \frac{1}{9}$ .

$a_{n} = \frac{1}{3} - \frac{1}{9} (n - 1)$

Step 5

Simplify each term.

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Step 5.1

Apply the distributive property.

$a_{n} = \frac{1}{3} - \frac{1}{9} n - \frac{1}{9} \cdot - 1$

Step 5.2

Combine $n$ and $\frac{1}{9}$ .

$a_{n} = \frac{1}{3} - \frac{n}{9} - \frac{1}{9} \cdot - 1$

Step 5.3

Multiply $- \frac{1}{9} \cdot - 1$ .

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Step 5.3.1

Multiply $- 1$ by $- 1$ .

$a_{n} = \frac{1}{3} - \frac{n}{9} + 1 (\frac{1}{9})$

Step 5.3.2

Multiply $\frac{1}{9}$ by $1$ .

$a_{n} = \frac{1}{3} - \frac{n}{9} + \frac{1}{9}$

Step 6

To write $\frac{1}{3}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a_{n} = - \frac{n}{9} + \frac{1}{3} \cdot \frac{3}{3} + \frac{1}{9}$

Step 7

Write each expression with a common denominator of $9$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

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

$a_{n} = - \frac{n}{9} + \frac{3}{3 \cdot 3} + \frac{1}{9}$

Step 7.2

Multiply $3$ by $3$ .

$a_{n} = - \frac{n}{9} + \frac{3}{9} + \frac{1}{9}$

Step 8

Combine the numerators over the common denominator.

$a_{n} = - \frac{n}{9} + \frac{3 + 1}{9}$

Step 9

Add $3$ and $1$ .

$a_{n} = - \frac{n}{9} + \frac{4}{9}$

Step 10

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

$a_{4} = - \frac{4}{9} + \frac{4}{9}$

Step 11

Combine the numerators over the common denominator.

$a_{4} = \frac{- 4 + 4}{9}$

Step 12

Simplify the expression.

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Step 12.1

Add $- 4$ and $4$ .

$a_{4} = \frac{0}{9}$

Step 12.2

Divide $0$ by $9$ .

$a_{4} = 0$