Finite Math Examples

$\frac{1 - {(1 + i)}^{- n}}{i}$

Step 1

Multiply the numerator and denominator of $\frac{1 - {(1 + i)}^{- n}}{i}$ by the conjugate of $i$ to make the denominator real.

$\frac{1 - {(1 + i)}^{- n}}{i} \cdot \frac{i}{i}$

Step 2

Multiply.

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Step 2.1

Combine.

$\frac{(1 - {(1 + i)}^{- n}) i}{i i}$

Step 2.2

Simplify the numerator.

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Step 2.2.1

Apply the distributive property.

$\frac{1 i - {(1 + i)}^{- n} i}{i i}$

Step 2.2.2

Multiply $i$ by $1$ .

$\frac{i - {(1 + i)}^{- n} i}{i i}$

Step 2.2.3

Reorder factors in $i - {(1 + i)}^{- n} i$ .

$\frac{i - i {(1 + i)}^{- n}}{i i}$

Step 2.3

Simplify the denominator.

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Step 2.3.1

Raise $i$ to the power of $1$ .

$\frac{i - i {(1 + i)}^{- n}}{i^{1} i}$

Step 2.3.2

Raise $i$ to the power of $1$ .

$\frac{i - i {(1 + i)}^{- n}}{i^{1} i^{1}}$

Step 2.3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{i - i {(1 + i)}^{- n}}{i^{1 + 1}}$

Step 2.3.4

Add $1$ and $1$ .

$\frac{i - i {(1 + i)}^{- n}}{i^{2}}$

Step 2.3.5

Rewrite $i^{2}$ as $- 1$ .

$\frac{i - i {(1 + i)}^{- n}}{- 1}$

Step 3

Simplify the expression.

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Step 3.1

Move the negative one from the denominator of $\frac{i - i {(1 + i)}^{- n}}{- 1}$ .

$- 1 \cdot (i - i {(1 + i)}^{- n})$

Step 3.2

Rewrite $- 1 \cdot (i - i {(1 + i)}^{- n})$ as $- (i - i {(1 + i)}^{- n})$ .

$- (i - i {(1 + i)}^{- n})$

Step 4

Apply the distributive property.

$- i - (- i {(1 + i)}^{- n})$

Step 5

Multiply $- (- i {(1 + i)}^{- n})$ .

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Step 5.1

Multiply $- 1$ by $- 1$ .

$- i + 1 (i {(1 + i)}^{- n})$

Step 5.2

Multiply ${(1 + i)}^{- n}$ by $1$ .

$- i + {(1 + i)}^{- n} i$

Step 6

Reorder factors in $- i + {(1 + i)}^{- n} i$ .

$- i + i {(1 + i)}^{- n}$