Algebra Examples

$\frac{2}{3}$ , $\frac{1}{6}$ , $- \frac{1}{3}$ , $- \frac{5}{6}$

Step 1

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

Arithmetic Sequence: $d = - \frac{1}{2}$

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 $d = - \frac{1}{2}$ .

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

Step 4

Simplify each term.

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Step 4.1

Apply the distributive property.

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

Step 4.2

Combine $n$ and $\frac{1}{2}$ .

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

Step 4.3

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

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Step 4.3.1

Multiply $- 1$ by $- 1$ .

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

Step 4.3.2

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

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

Step 5

To write $\frac{2}{3}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 6

To write $\frac{1}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 7

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 7.1

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

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

Step 7.2

Multiply $3$ by $2$ .

$a_{n} = - \frac{n}{2} + \frac{2 \cdot 2}{6} + \frac{1}{2} \cdot \frac{3}{3}$

Step 7.3

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

$a_{n} = - \frac{n}{2} + \frac{2 \cdot 2}{6} + \frac{3}{2 \cdot 3}$

Step 7.4

Multiply $2$ by $3$ .

$a_{n} = - \frac{n}{2} + \frac{2 \cdot 2}{6} + \frac{3}{6}$

Step 8

Combine the numerators over the common denominator.

$a_{n} = - \frac{n}{2} + \frac{2 \cdot 2 + 3}{6}$

Step 9

Simplify the numerator.

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Step 9.1

Multiply $2$ by $2$ .

$a_{n} = - \frac{n}{2} + \frac{4 + 3}{6}$

Step 9.2

Add $4$ and $3$ .

$a_{n} = - \frac{n}{2} + \frac{7}{6}$

Step 10

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

$a_{5} = - \frac{5}{2} + \frac{7}{6}$

Step 11

To write $- \frac{5}{2}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$a_{5} = - \frac{5}{2} \cdot \frac{3}{3} + \frac{7}{6}$

Step 12

Write each expression with a common denominator of $6$ , by multiplying each by an appropriate factor of $1$ .

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Step 12.1

Multiply $\frac{5}{2}$ by $\frac{3}{3}$ .

$a_{5} = - \frac{5 \cdot 3}{2 \cdot 3} + \frac{7}{6}$

Step 12.2

Multiply $2$ by $3$ .

$a_{5} = - \frac{5 \cdot 3}{6} + \frac{7}{6}$

Step 13

Combine the numerators over the common denominator.

$a_{5} = \frac{- 5 \cdot 3 + 7}{6}$

Step 14

Simplify the numerator.

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Step 14.1

Multiply $- 5$ by $3$ .

$a_{5} = \frac{- 15 + 7}{6}$

Step 14.2

Add $- 15$ and $7$ .

$a_{5} = \frac{- 8}{6}$

Step 15

Cancel the common factor of $- 8$ and $6$ .

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Step 15.1

Factor $2$ out of $- 8$ .

$a_{5} = \frac{2 (- 4)}{6}$

Step 15.2

Cancel the common factors.

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Step 15.2.1

Factor $2$ out of $6$ .

$a_{5} = \frac{2 \cdot - 4}{2 \cdot 3}$

Step 15.2.2

Cancel the common factor.

$a_{5} = \frac{2 \cdot - 4}{2 \cdot 3}$

Step 15.2.3

Rewrite the expression.

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

Step 16

Move the negative in front of the fraction.

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