Algebra Examples

$- 32$ , $10 \frac{2}{3}$ , $- 3 \frac{5}{9}$ , $1 \frac{5}{27}$

Step 1

Convert $10 \frac{2}{3}$ to an improper fraction.

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Step 1.1

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

$- 32, 10 + \frac{2}{3}, - 3 \frac{5}{9}, 1 \frac{5}{27}$

Step 1.2

Add $10$ and $\frac{2}{3}$ .

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Step 1.2.1

To write $10$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- 32, 10 \cdot \frac{3}{3} + \frac{2}{3}, - 3 \frac{5}{9}, 1 \frac{5}{27}$

Step 1.2.2

Combine $10$ and $\frac{3}{3}$ .

$- 32, \frac{10 \cdot 3}{3} + \frac{2}{3}, - 3 \frac{5}{9}, 1 \frac{5}{27}$

Step 1.2.3

Combine the numerators over the common denominator.

$- 32, \frac{10 \cdot 3 + 2}{3}, - 3 \frac{5}{9}, 1 \frac{5}{27}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $10$ by $3$ .

$- 32, \frac{30 + 2}{3}, - 3 \frac{5}{9}, 1 \frac{5}{27}$

Step 1.2.4.2

Add $30$ and $2$ .

$- 32, \frac{32}{3}, - 3 \frac{5}{9}, 1 \frac{5}{27}$

Step 2

Convert $3 \frac{5}{9}$ to an improper fraction.

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Step 2.1

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

$- 32, \frac{32}{3}, - (3 + \frac{5}{9}), 1 \frac{5}{27}$

Step 2.2

Add $3$ and $\frac{5}{9}$ .

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Step 2.2.1

To write $3$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$- 32, \frac{32}{3}, - (3 \cdot \frac{9}{9} + \frac{5}{9}), 1 \frac{5}{27}$

Step 2.2.2

Combine $3$ and $\frac{9}{9}$ .

$- 32, \frac{32}{3}, - (\frac{3 \cdot 9}{9} + \frac{5}{9}), 1 \frac{5}{27}$

Step 2.2.3

Combine the numerators over the common denominator.

$- 32, \frac{32}{3}, - \frac{3 \cdot 9 + 5}{9}, 1 \frac{5}{27}$

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Multiply $3$ by $9$ .

$- 32, \frac{32}{3}, - \frac{27 + 5}{9}, 1 \frac{5}{27}$

Step 2.2.4.2

Add $27$ and $5$ .

$- 32, \frac{32}{3}, - \frac{32}{9}, 1 \frac{5}{27}$

Step 3

Convert $1 \frac{5}{27}$ to an improper fraction.

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Step 3.1

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

$- 32, \frac{32}{3}, - \frac{32}{9}, 1 + \frac{5}{27}$

Step 3.2

Add $1$ and $\frac{5}{27}$ .

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Step 3.2.1

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

$- 32, \frac{32}{3}, - \frac{32}{9}, \frac{27}{27} + \frac{5}{27}$

Step 3.2.2

Combine the numerators over the common denominator.

$- 32, \frac{32}{3}, - \frac{32}{9}, \frac{27 + 5}{27}$

Step 3.2.3

Add $27$ and $5$ .

$- 32, \frac{32}{3}, - \frac{32}{9}, \frac{32}{27}$

Step 4

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $- \frac{1}{3}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = - \frac{1}{3}$

Step 5

This is the form of a geometric sequence.

$a_{n} = a_{1} r^{n - 1}$

Step 6

Substitute in the values of $a_{1} = - 32$ and $r = - \frac{1}{3}$ .

$a_{n} = - 32 {(- \frac{1}{3})}^{n - 1}$

Step 7

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 7.1

Apply the product rule to $- \frac{1}{3}$ .

$a_{n} = - 32 ({(- 1)}^{n - 1} {(\frac{1}{3})}^{n - 1})$

Step 7.2

Apply the product rule to $\frac{1}{3}$ .

$a_{n} = - 32 ({(- 1)}^{n - 1} \frac{1^{n - 1}}{3^{n - 1}})$

Step 8

One to any power is one.

$a_{n} = - 32 ({(- 1)}^{n - 1} \frac{1}{3^{n - 1}})$

Step 9

Combine ${(- 1)}^{n - 1}$ and $\frac{1}{3^{n - 1}}$ .

$a_{n} = - 32 \frac{{(- 1)}^{n - 1}}{3^{n - 1}}$

Step 10

Combine $- 32$ and $\frac{{(- 1)}^{n - 1}}{3^{n - 1}}$ .

$a_{n} = \frac{- 32 {(- 1)}^{n - 1}}{3^{n - 1}}$

Step 11

Move the negative in front of the fraction.

$a_{n} = - \frac{32 {(- 1)}^{n - 1}}{3^{n - 1}}$