Calculus Examples

$\sum_{i = 1}^{7} {(- 3)}^{i}$

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

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Step 2.1

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

$r = \frac{{(- 3)}^{i + 1}}{{(- 3)}^{i}}$

Step 2.2

Cancel the common factor of ${(- 3)}^{i + 1}$ and ${(- 3)}^{i}$ .

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Step 2.2.1

Factor ${(- 3)}^{i}$ out of ${(- 3)}^{i + 1}$ .

$r = \frac{{(- 3)}^{i} \cdot - 3}{{(- 3)}^{i}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

$r = \frac{{(- 3)}^{i} \cdot - 3}{{(- 3)}^{i} \cdot 1}$

Step 2.2.2.2

Cancel the common factor.

$r = \frac{{(- 3)}^{i} \cdot - 3}{{(- 3)}^{i} \cdot 1}$

Step 2.2.2.3

Rewrite the expression.

$r = \frac{- 3}{1}$

Step 2.2.2.4

Divide $- 3$ by $1$ .

$r = - 3$

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 $i$ into ${(- 3)}^{i}$ .

$a = {(- 3)}^{1}$

Step 3.2

Evaluate the exponent.

$a = - 3$

Step 4

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

$- 3 \frac{1 - {(- 3)}^{7}}{1 - - 3}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 3$ to the power of $7$ .

$- 3 \frac{1 - - 2187}{1 - - 3}$

Step 5.1.2

Multiply $- 1$ by $- 2187$ .

$- 3 \frac{1 + 2187}{1 - - 3}$

Step 5.1.3

Add $1$ and $2187$ .

$- 3 \frac{2188}{1 - - 3}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $- 3$ .

$- 3 \frac{2188}{1 + 3}$

Step 5.2.2

Add $1$ and $3$ .

$- 3 (\frac{2188}{4})$

Step 5.3

Divide $2188$ by $4$ .

$- 3 \cdot 547$

Step 5.4

Multiply $- 3$ by $547$ .

$- 1641$