Algebra Examples

$2$ , $1 \frac{8}{9}$ , $1 \frac{7}{9}$ , $1 \frac{2}{3}$

Step 1

Convert $1 \frac{8}{9}$ to an improper fraction.

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Step 1.1

A mixed number is an addition of its whole and fractional parts.

$2, 1 + \frac{8}{9}, 1 \frac{7}{9}, 1 \frac{2}{3}$

Step 1.2

Add $1$ and $\frac{8}{9}$ .

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Step 1.2.1

Write $1$ as a fraction with a common denominator.

$2, \frac{9}{9} + \frac{8}{9}, 1 \frac{7}{9}, 1 \frac{2}{3}$

Step 1.2.2

Combine the numerators over the common denominator.

$2, \frac{9 + 8}{9}, 1 \frac{7}{9}, 1 \frac{2}{3}$

Step 1.2.3

Add $9$ and $8$ .

$2, \frac{17}{9}, 1 \frac{7}{9}, 1 \frac{2}{3}$

Step 2

Convert $1 \frac{7}{9}$ to an improper fraction.

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Step 2.1

A mixed number is an addition of its whole and fractional parts.

$2, \frac{17}{9}, 1 + \frac{7}{9}, 1 \frac{2}{3}$

Step 2.2

Add $1$ and $\frac{7}{9}$ .

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Step 2.2.1

Write $1$ as a fraction with a common denominator.

$2, \frac{17}{9}, \frac{9}{9} + \frac{7}{9}, 1 \frac{2}{3}$

Step 2.2.2

Combine the numerators over the common denominator.

$2, \frac{17}{9}, \frac{9 + 7}{9}, 1 \frac{2}{3}$

Step 2.2.3

Add $9$ and $7$ .

$2, \frac{17}{9}, \frac{16}{9}, 1 \frac{2}{3}$

Step 3

Convert $1 \frac{2}{3}$ to an improper fraction.

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Step 3.1

A mixed number is an addition of its whole and fractional parts.

$2, \frac{17}{9}, \frac{16}{9}, 1 + \frac{2}{3}$

Step 3.2

Add $1$ and $\frac{2}{3}$ .

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Step 3.2.1

Write $1$ as a fraction with a common denominator.

$2, \frac{17}{9}, \frac{16}{9}, \frac{3}{3} + \frac{2}{3}$

Step 3.2.2

Combine the numerators over the common denominator.

$2, \frac{17}{9}, \frac{16}{9}, \frac{3 + 2}{3}$

Step 3.2.3

Add $3$ and $2$ .

$2, \frac{17}{9}, \frac{16}{9}, \frac{5}{3}$

Step 4

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 5

This is the formula of an arithmetic sequence.

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

Step 6

Substitute in the values of $a_{1} = 2$ and $d = - \frac{1}{9}$ .

$a_{n} = 2 - \frac{1}{9} (n - 1)$

Step 7

Simplify each term.

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Step 7.1

Apply the distributive property.

$a_{n} = 2 - \frac{1}{9} n - \frac{1}{9} \cdot - 1$

Step 7.2

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

$a_{n} = 2 - \frac{n}{9} - \frac{1}{9} \cdot - 1$

Step 7.3

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

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Step 7.3.1

Multiply $- 1$ by $- 1$ .

$a_{n} = 2 - \frac{n}{9} + 1 (\frac{1}{9})$

Step 7.3.2

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

$a_{n} = 2 - \frac{n}{9} + \frac{1}{9}$

Step 8

To write $2$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$a_{n} = - \frac{n}{9} + 2 \cdot \frac{9}{9} + \frac{1}{9}$

Step 9

Combine $2$ and $\frac{9}{9}$ .

$a_{n} = - \frac{n}{9} + \frac{2 \cdot 9}{9} + \frac{1}{9}$

Step 10

Combine the numerators over the common denominator.

$a_{n} = - \frac{n}{9} + \frac{2 \cdot 9 + 1}{9}$

Step 11

Simplify the numerator.

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Step 11.1

Multiply $2$ by $9$ .

$a_{n} = - \frac{n}{9} + \frac{18 + 1}{9}$

Step 11.2

Add $18$ and $1$ .

$a_{n} = - \frac{n}{9} + \frac{19}{9}$