Precalculus Examples

$\sum_{i = 1}^{11} 3 \cdot 2^{i}$

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 $r$ .

$r = \frac{3 \cdot 2^{i + 1}}{3 \cdot 2^{i}}$

Step 2.2

Simplify.

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Step 2.2.1

Cancel the common factor of $3$ .

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Step 2.2.1.1

Cancel the common factor.

$r = \frac{3 \cdot 2^{i + 1}}{3 \cdot 2^{i}}$

Step 2.2.1.2

Rewrite the expression.

$r = \frac{2^{i + 1}}{2^{i}}$

Step 2.2.2

Cancel the common factor of $2^{i + 1}$ and $2^{i}$ .

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Step 2.2.2.1

Factor $2^{i}$ out of $2^{i + 1}$ .

$r = \frac{2^{i} \cdot 2}{2^{i}}$

Step 2.2.2.2

Cancel the common factors.

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Step 2.2.2.2.1

Multiply by $1$ .

$r = \frac{2^{i} \cdot 2}{2^{i} \cdot 1}$

Step 2.2.2.2.2

Cancel the common factor.

$r = \frac{2^{i} \cdot 2}{2^{i} \cdot 1}$

Step 2.2.2.2.3

Rewrite the expression.

$r = \frac{2}{1}$

Step 2.2.2.2.4

Divide $2$ by $1$ .

$r = 2$

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 $i$ into $3 \cdot 2^{i}$ .

$a = 3 \cdot 2^{1}$

Step 3.2

Simplify.

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Step 3.2.1

Evaluate the exponent.

$a = 3 \cdot 2$

Step 3.2.2

Multiply $3$ by $2$ .

$a = 6$

Step 4

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

$6 \frac{1 - 2^{11}}{1 - 1 \cdot 2}$

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 $11$ .

$6 \frac{1 - 1 \cdot 2048}{1 - 1 \cdot 2}$

Step 5.1.2

Multiply $- 1$ by $2048$ .

$6 \frac{1 - 2048}{1 - 1 \cdot 2}$

Step 5.1.3

Subtract $2048$ from $1$ .

$6 \frac{- 2047}{1 - 1 \cdot 2}$

Step 5.2

Simplify the denominator.

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Step 5.2.1

Multiply $- 1$ by $2$ .

$6 \frac{- 2047}{1 - 2}$

Step 5.2.2

Subtract $2$ from $1$ .

$6 (\frac{- 2047}{- 1})$

Step 5.3

Divide $- 2047$ by $- 1$ .

$6 \cdot 2047$

Step 5.4

Multiply $6$ by $2047$ .

$12282$