Calculus Examples

$\sum_{n = 1}^{7} 2 {(- 2)}^{n - 1}$

Step 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.

Step 2

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

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Step 2.1

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

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

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $2$ .

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Step 2.2.1.1

Cancel the common factor.

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

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

Cancel the common factor of ${(- 2)}^{n + 1 - 1}$ and ${(- 2)}^{n - 1}$ .

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Step 2.2.2.1

Factor ${(- 2)}^{n - 1}$ out of ${(- 2)}^{n + 1 - 1}$ .

$r = \frac{{(- 2)}^{n - 1} {(- 2)}^{n + 0 - (n - 1)}}{{(- 2)}^{n - 1}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{{(- 2)}^{n + 0 - (n - 1)}}{1}$

Step 2.2.2.2.4

Divide ${(- 2)}^{n + 0 - (n - 1)}$ by $1$ .

$r = {(- 2)}^{n + 0 - (n - 1)}$

Step 2.2.3

Add $n$ and $0$ .

$r = {(- 2)}^{n - (n - 1)}$

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

$r = {(- 2)}^{n - n - - 1}$

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = {(- 2)}^{n - n + 1}$

Step 2.2.5

Subtract $n$ from $n$ .

$r = {(- 2)}^{0 + 1}$

Step 2.2.6

Add $0$ and $1$ .

$r = {(- 2)}^{1}$

Step 2.2.7

Evaluate the exponent.

$r = - 2$

Step 3

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

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Step 3.1

Substitute $1$ for $n$ into $2 {(- 2)}^{n - 1}$ .

$a = 2 {(- 2)}^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = 2 {(- 2)}^{0}$

Step 3.2.2

Anything raised to $0$ is $1$ .

$a = 2 \cdot 1$

Step 3.2.3

Multiply $2$ by $1$ .

$a = 2$

Step 4

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

$2 \frac{1 - {(- 2)}^{7}}{1 - - 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 2$ to the power of $7$ .

$2 \frac{1 - - 128}{1 - - 2}$

Step 5.1.2

Multiply $- 1$ by $- 128$ .

$2 \frac{1 + 128}{1 - - 2}$

Step 5.1.3

Add $1$ and $128$ .

$2 \frac{129}{1 - - 2}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $- 2$ .

$2 \frac{129}{1 + 2}$

Step 5.2.2

Add $1$ and $2$ .

$2 (\frac{129}{3})$

Step 5.3

Divide $129$ by $3$ .

$2 \cdot 43$

Step 5.4

Multiply $2$ by $43$ .

$86$