Precalculus Examples

$97$ , $48 \frac{1}{2}$ , $24 \frac{1}{4}$ , $12 \frac{1}{8}$

Step 1

Convert $48 \frac{1}{2}$ to an improper fraction.

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Step 1.1

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

$97, 48 + \frac{1}{2}, 24 \frac{1}{4}, 12 \frac{1}{8}$

Step 1.2

Add $48$ and $\frac{1}{2}$ .

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Step 1.2.1

To write $48$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$97, 48 \cdot \frac{2}{2} + \frac{1}{2}, 24 \frac{1}{4}, 12 \frac{1}{8}$

Step 1.2.2

Combine $48$ and $\frac{2}{2}$ .

$97, \frac{48 \cdot 2}{2} + \frac{1}{2}, 24 \frac{1}{4}, 12 \frac{1}{8}$

Step 1.2.3

Combine the numerators over the common denominator.

$97, \frac{48 \cdot 2 + 1}{2}, 24 \frac{1}{4}, 12 \frac{1}{8}$

Step 1.2.4

Simplify the numerator.

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Step 1.2.4.1

Multiply $48$ by $2$ .

$97, \frac{96 + 1}{2}, 24 \frac{1}{4}, 12 \frac{1}{8}$

Step 1.2.4.2

Add $96$ and $1$ .

$97, \frac{97}{2}, 24 \frac{1}{4}, 12 \frac{1}{8}$

Step 2

Convert $24 \frac{1}{4}$ to an improper fraction.

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Step 2.1

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

$97, \frac{97}{2}, 24 + \frac{1}{4}, 12 \frac{1}{8}$

Step 2.2

Add $24$ and $\frac{1}{4}$ .

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Step 2.2.1

To write $24$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$97, \frac{97}{2}, 24 \cdot \frac{4}{4} + \frac{1}{4}, 12 \frac{1}{8}$

Step 2.2.2

Combine $24$ and $\frac{4}{4}$ .

$97, \frac{97}{2}, \frac{24 \cdot 4}{4} + \frac{1}{4}, 12 \frac{1}{8}$

Step 2.2.3

Combine the numerators over the common denominator.

$97, \frac{97}{2}, \frac{24 \cdot 4 + 1}{4}, 12 \frac{1}{8}$

Step 2.2.4

Simplify the numerator.

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Step 2.2.4.1

Multiply $24$ by $4$ .

$97, \frac{97}{2}, \frac{96 + 1}{4}, 12 \frac{1}{8}$

Step 2.2.4.2

Add $96$ and $1$ .

$97, \frac{97}{2}, \frac{97}{4}, 12 \frac{1}{8}$

Step 3

Convert $12 \frac{1}{8}$ to an improper fraction.

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Step 3.1

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

$97, \frac{97}{2}, \frac{97}{4}, 12 + \frac{1}{8}$

Step 3.2

Add $12$ and $\frac{1}{8}$ .

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Step 3.2.1

To write $12$ as a fraction with a common denominator, multiply by $\frac{8}{8}$ .

$97, \frac{97}{2}, \frac{97}{4}, 12 \cdot \frac{8}{8} + \frac{1}{8}$

Step 3.2.2

Combine $12$ and $\frac{8}{8}$ .

$97, \frac{97}{2}, \frac{97}{4}, \frac{12 \cdot 8}{8} + \frac{1}{8}$

Step 3.2.3

Combine the numerators over the common denominator.

$97, \frac{97}{2}, \frac{97}{4}, \frac{12 \cdot 8 + 1}{8}$

Step 3.2.4

Simplify the numerator.

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Step 3.2.4.1

Multiply $12$ by $8$ .

$97, \frac{97}{2}, \frac{97}{4}, \frac{96 + 1}{8}$

Step 3.2.4.2

Add $96$ and $1$ .

$97, \frac{97}{2}, \frac{97}{4}, \frac{97}{8}$

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 $\frac{1}{2}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{1}{2}$

Step 5

This is the form of a geometric sequence.

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

Step 6

Substitute in the values of $a_{1} = 97$ and $r = \frac{1}{2}$ .

$a_{n} = 97 {(\frac{1}{2})}^{n - 1}$

Step 7

Apply the product rule to $\frac{1}{2}$ .

$a_{n} = 97 \frac{1^{n - 1}}{2^{n - 1}}$

Step 8

One to any power is one.

$a_{n} = 97 \frac{1}{2^{n - 1}}$

Step 9

Combine $97$ and $\frac{1}{2^{n - 1}}$ .

$a_{n} = \frac{97}{2^{n - 1}}$