Calculus Examples

$\sum_{i = 1}^{9} 4^{i}$

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 $r$ .

$r = \frac{4^{i + 1}}{4^{i}}$

Step 2.2

Cancel the common factor of $4^{i + 1}$ and $4^{i}$ .

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Step 2.2.1

Factor $4^{i}$ out of $4^{i + 1}$ .

$r = \frac{4^{i} \cdot 4}{4^{i}}$

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

$r = \frac{4^{i} \cdot 4}{4^{i} \cdot 1}$

Step 2.2.2.2

Cancel the common factor.

$r = \frac{4^{i} \cdot 4}{4^{i} \cdot 1}$

Step 2.2.2.3

Rewrite the expression.

$r = \frac{4}{1}$

Step 2.2.2.4

Divide $4$ by $1$ .

$r = 4$

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 $4^{i}$ .

$a = 4^{1}$

Step 3.2

Evaluate the exponent.

$a = 4$

Step 4

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

$4 \frac{1 - 4^{9}}{1 - 1 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $4$ to the power of $9$ .

$4 \frac{1 - 1 \cdot 262144}{1 - 1 \cdot 4}$

Step 5.1.2

Multiply $- 1$ by $262144$ .

$4 \frac{1 - 262144}{1 - 1 \cdot 4}$

Step 5.1.3

Subtract $262144$ from $1$ .

$4 \frac{- 262143}{1 - 1 \cdot 4}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $4$ .

$4 \frac{- 262143}{1 - 4}$

Step 5.2.2

Subtract $4$ from $1$ .

$4 (\frac{- 262143}{- 3})$

Step 5.3

Divide $- 262143$ by $- 3$ .

$4 \cdot 87381$

Step 5.4

Multiply $4$ by $87381$ .

$349524$