Precalculus Examples

$\sum_{k = 5}^{15} (- 10) {(- 1)}^{k - 1}$

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 $r = \frac{a_{k + 1}}{a_{k}}$ and simplifying.

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Step 2.1

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

$r = \frac{- 10 {(- 1)}^{k + 1 - 1}}{- 10 {(- 1)}^{k - 1}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $- 10$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{- 10 {(- 1)}^{k + 1 - 1}}{- 10 {(- 1)}^{k - 1}}$

Step 2.2.1.2

Rewrite the expression.

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

Step 2.2.2

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

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Step 2.2.2.1

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

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

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2.2

Cancel the common factor.

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

Step 2.2.2.2.3

Rewrite the expression.

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

Step 2.2.2.2.4

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

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

Step 2.2.3

Add $k$ and $0$ .

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

Step 2.2.4

Simplify each term.

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Step 2.2.4.1

Apply the distributive property.

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

Step 2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = {(- 1)}^{k - k + 1}$

Step 2.2.5

Subtract $k$ from $k$ .

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

Step 2.2.6

Add $0$ and $1$ .

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

Step 2.2.7

Evaluate the exponent.

$r = - 1$

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 $5$ for $k$ into $(- 10) {(- 1)}^{k - 1}$ .

$a = - 10 {(- 1)}^{5 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $5$ .

$a = - 10 {(- 1)}^{4}$

Step 3.2.2

Raise $- 1$ to the power of $4$ .

$a = - 10 \cdot 1$

Step 3.2.3

Multiply $- 10$ by $1$ .

$a = - 10$

Step 4

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

$- 10 \frac{1 - {(- 1)}^{11}}{1 - - 1}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Multiply $- 1$ by ${(- 1)}^{11}$ by adding the exponents.

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Step 5.1.1.1

Multiply $- 1$ by ${(- 1)}^{11}$ .

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Step 5.1.1.1.1

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

$- 10 \frac{1 + {(- 1)}^{1} {(- 1)}^{11}}{1 - - 1}$

Step 5.1.1.1.2

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

$- 10 \frac{1 + {(- 1)}^{1 + 11}}{1 - - 1}$

Step 5.1.1.2

Add $1$ and $11$ .

$- 10 \frac{1 + {(- 1)}^{12}}{1 - - 1}$

Step 5.1.2

Raise $- 1$ to the power of $12$ .

$- 10 \frac{1 + 1}{1 - - 1}$

Step 5.1.3

Add $1$ and $1$ .

$- 10 \frac{2}{1 - - 1}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $- 1$ .

$- 10 \frac{2}{1 + 1}$

Step 5.2.2

Add $1$ and $1$ .

$- 10 (\frac{2}{2})$

Step 5.3

Cancel the common factor of $2$ .

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Step 5.3.1

Cancel the common factor.

$- 10 (\frac{2}{2})$

Step 5.3.2

Rewrite the expression.

$- 10 \cdot 1$

Step 5.4

Multiply $- 10$ by $1$ .

$- 10$