Finite Math Examples

a (n) = \frac{1}{3} \cdot (1 - {(- \frac{1}{2})}^{n - 1})

$a (n) = \frac{1}{3} \cdot (1 - {(- \frac{1}{2})}^{n - 1})$

Step 1

Simplify.

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Step 1.1

Simplify each term.

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Step 1.1.1

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

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

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Step 1.1.1.1

Apply the product rule to

- \frac{1}{2}

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

a n = \frac{1}{3} \cdot (1 - ({(- 1)}^{n - 1} {(\frac{1}{2})}^{n - 1}))

$a n = \frac{1}{3} \cdot (1 - ({(- 1)}^{n - 1} {(\frac{1}{2})}^{n - 1}))$

Step 1.1.1.2

Apply the product rule to

\frac{1}{2}

$\frac{1}{2}$ .

a n = \frac{1}{3} \cdot (1 - ({(- 1)}^{n - 1} \frac{1^{n - 1}}{2^{n - 1}}))

$a n = \frac{1}{3} \cdot (1 - ({(- 1)}^{n - 1} \frac{1^{n - 1}}{2^{n - 1}}))$

a n = \frac{1}{3} \cdot (1 - ({(- 1)}^{n - 1} \frac{1^{n - 1}}{2^{n - 1}}))

$a n = \frac{1}{3} \cdot (1 - ({(- 1)}^{n - 1} \frac{1^{n - 1}}{2^{n - 1}}))$

Step 1.1.2

Multiply

- 1

$- 1$ by

{(- 1)}^{n - 1}

${(- 1)}^{n - 1}$ by adding the exponents.

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Step 1.1.2.1

Move

{(- 1)}^{n - 1}

${(- 1)}^{n - 1}$ .

a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n - 1} \cdot - 1 \frac{1^{n - 1}}{2^{n - 1}})

$a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n - 1} \cdot - 1 \frac{1^{n - 1}}{2^{n - 1}})$

Step 1.1.2.2

Multiply

{(- 1)}^{n - 1}

${(- 1)}^{n - 1}$ by

- 1

$- 1$ .

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Step 1.1.2.2.1

Raise

- 1

$- 1$ to the power of

1

$1$ .

a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n - 1} \cdot {(- 1)}^{1} \frac{1^{n - 1}}{2^{n - 1}})

$a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n - 1} \cdot {(- 1)}^{1} \frac{1^{n - 1}}{2^{n - 1}})$

Step 1.1.2.2.2

Use the power rule

a^{m} a^{n} = a^{m + n}

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n - 1 + 1} \frac{1^{n - 1}}{2^{n - 1}})

$a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n - 1 + 1} \frac{1^{n - 1}}{2^{n - 1}})$

a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n - 1 + 1} \frac{1^{n - 1}}{2^{n - 1}})

$a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n - 1 + 1} \frac{1^{n - 1}}{2^{n - 1}})$

Step 1.1.2.3

Combine the opposite terms in

n - 1 + 1

$n - 1 + 1$ .

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Step 1.1.2.3.1

Add

$- 1$ and

$1$ .

$a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n + 0} \frac{1^{n - 1}}{2^{n - 1}})$

Step 1.1.2.3.2

Add

$n$ and

$0$ .

$a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n} \frac{1^{n - 1}}{2^{n - 1}})$

Step 1.1.3

One to any power is one.

$a n = \frac{1}{3} \cdot (1 + {(- 1)}^{n} \frac{1}{2^{n - 1}})$

Step 1.1.4

Combine

${(- 1)}^{n}$ and

$\frac{1}{2^{n - 1}}$ .

$a n = \frac{1}{3} \cdot (1 + \frac{{(- 1)}^{n}}{2^{n - 1}})$

Step 1.2

Apply the distributive property.

$a n = \frac{1}{3} \cdot 1 + \frac{1}{3} \cdot \frac{{(- 1)}^{n}}{2^{n - 1}}$

Step 1.3

Multiply

$\frac{1}{3}$ by

$1$ .

$a n = \frac{1}{3} + \frac{1}{3} \cdot \frac{{(- 1)}^{n}}{2^{n - 1}}$

Step 1.4

Combine.

$a n = \frac{1}{3} + \frac{1 {(- 1)}^{n}}{3 \cdot 2^{n - 1}}$

Step 1.5

Multiply

${(- 1)}^{n}$ by

$1$ .

$a n = \frac{1}{3} + \frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}$

Step 2

Divide each term in

$a n = \frac{1}{3} + \frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}$ by

$n$ and simplify.

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Step 2.1

Divide each term in

$a n = \frac{1}{3} + \frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}$ by

$n$ .

$\frac{a n}{n} = \frac{\frac{1}{3}}{n} + \frac{\frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}}{n}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Cancel the common factor of

$n$ .

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Step 2.2.1.1

Cancel the common factor.

$\frac{a n}{n} = \frac{\frac{1}{3}}{n} + \frac{\frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}}{n}$

Step 2.2.1.2

Divide

$a$ by

$1$ .

$a = \frac{\frac{1}{3}}{n} + \frac{\frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}}{n}$

Step 2.3

Simplify the right side.

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Step 2.3.1

Combine the numerators over the common denominator.

$a = \frac{\frac{1}{3} + \frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}}{n}$

Step 2.3.2

Simplify the numerator.

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Step 2.3.2.1

To write

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

$\frac{2^{n - 1}}{2^{n - 1}}$ .

$a = \frac{\frac{1}{3} \cdot \frac{2^{n - 1}}{2^{n - 1}} + \frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}}{n}$

Step 2.3.2.2

Multiply

$\frac{1}{3}$ by

$\frac{2^{n - 1}}{2^{n - 1}}$ .

$a = \frac{\frac{2^{n - 1}}{3 \cdot 2^{n - 1}} + \frac{{(- 1)}^{n}}{3 \cdot 2^{n - 1}}}{n}$

Step 2.3.2.3

Combine the numerators over the common denominator.

$a = \frac{\frac{2^{n - 1} + {(- 1)}^{n}}{3 \cdot 2^{n - 1}}}{n}$

Step 2.3.3

Multiply the numerator by the reciprocal of the denominator.

$a = \frac{2^{n - 1} + {(- 1)}^{n}}{3 \cdot 2^{n - 1}} \cdot \frac{1}{n}$

Step 2.3.4

Multiply

$\frac{2^{n - 1} + {(- 1)}^{n}}{3 \cdot 2^{n - 1}}$ by

$\frac{1}{n}$ .

$a = \frac{2^{n - 1} + {(- 1)}^{n}}{3 \cdot 2^{n - 1} n}$

Step 2.3.5

Reorder factors in

$\frac{2^{n - 1} + {(- 1)}^{n}}{3 \cdot 2^{n - 1} n}$ .

$a = \frac{2^{n - 1} + {(- 1)}^{n}}{3 n \cdot 2^{n - 1}}$