Calculus Examples

$\sum_{n = 1}^{4} 100 {(- 4)}^{n - 1}$

The sum of a finite geometric series can be found using the formula $a (\frac{1 - r^{n}}{1 - r})$ where $a$ is the first term and $r$ is the ratio between successive terms.

Find the ratio of successive terms by plugging into the formula $r = \frac{a_{n + 1}}{a_{n}}$ and simplifying.

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Substitute $a_{n}$ and $a_{n + 1}$ into the formula for $r$ .

$r = \frac{100 {(- 4)}^{n + 1 - 1}}{100 {(- 4)}^{n - 1}}$

Simplify.

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Cancel the common factor of $100$ .

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Cancel the common factor.

$r = \frac{100 {(- 4)}^{n + 1 - 1}}{100 {(- 4)}^{n - 1}}$

Rewrite the expression.

$r = \frac{{(- 4)}^{n + 1 - 1}}{{(- 4)}^{n - 1}}$

Cancel the common factor of ${(- 4)}^{n + 1 - 1}$ and ${(- 4)}^{n - 1}$ .

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Factor ${(- 4)}^{n - 1}$ out of ${(- 4)}^{n + 1 - 1}$ .

$r = \frac{{(- 4)}^{n - 1} {(- 4)}^{n + 0 - (n - 1)}}{{(- 4)}^{n - 1}}$

Cancel the common factors.

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Multiply by $1$ .

$r = \frac{{(- 4)}^{n - 1} {(- 4)}^{n + 0 - (n - 1)}}{{(- 4)}^{n - 1} \cdot 1}$

Cancel the common factor.

$r = \frac{{(- 4)}^{n - 1} {(- 4)}^{n + 0 - (n - 1)}}{{(- 4)}^{n - 1} \cdot 1}$

Rewrite the expression.

$r = \frac{{(- 4)}^{n + 0 - (n - 1)}}{1}$

Divide ${(- 4)}^{n + 0 - (n - 1)}$ by $1$ .

$r = {(- 4)}^{n + 0 - (n - 1)}$

Add $n$ and $0$ .

$r = {(- 4)}^{n - (n - 1)}$

Simplify each term.

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Apply the distributive property.

$r = {(- 4)}^{n - n - - 1}$

Multiply $- 1$ by $- 1$ .

$r = {(- 4)}^{n - n + 1}$

Subtract $n$ from $n$ .

$r = {(- 4)}^{0 + 1}$

Add $0$ and $1$ .

$r = {(- 4)}^{1}$

Evaluate the exponent.

$r = - 4$

Find the first term in the series by substituting in the lower bound and simplifying.

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Substitute $1$ for $n$ into $100 {(- 4)}^{n - 1}$ .

$a = 100 {(- 4)}^{1 - 1}$

Simplify.

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Subtract $1$ from $1$ .

$a = 100 {(- 4)}^{0}$

Anything raised to $0$ is $1$ .

$a = 100 \cdot 1$

Multiply $100$ by $1$ .

$a = 100$

Substitute the values of the ratio, first term, and number of terms into the sum formula.

$100 \frac{1 - {(- 4)}^{4}}{1 - - 4}$

Simplify.

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Simplify the numerator.

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Raise $- 4$ to the power of $4$ .

$100 \frac{1 - 1 \cdot 256}{1 - - 4}$

Multiply $- 1$ by $256$ .

$100 \frac{1 - 256}{1 - - 4}$

Subtract $256$ from $1$ .

$100 \frac{- 255}{1 - - 4}$

Simplify the denominator.

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Multiply $- 1$ by $- 4$ .

$100 \frac{- 255}{1 + 4}$

Add $1$ and $4$ .

$100 (\frac{- 255}{5})$

Cancel the common factor of $5$ .

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Factor $5$ out of $100$ .

$5 (20) \frac{- 255}{5}$

Cancel the common factor.

$5 \cdot 20 \frac{- 255}{5}$

Rewrite the expression.

$20 \cdot - 255$

Multiply $20$ by $- 255$ .

$- 5100$