Calculus Examples

$\sum_{x = 1}^{10} 6 {(2)}^{x}$

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.

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

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

$r = \frac{6 \cdot 2^{x + 1}}{6 \cdot 2^{x}}$

Simplify.

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Cancel the common factor of $6$ .

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Cancel the common factor.

$r = \frac{6 \cdot 2^{x + 1}}{6 \cdot 2^{x}}$

Rewrite the expression.

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

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

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

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

Cancel the common factors.

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Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2$ by $1$ .

$r = 2$

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

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Substitute $1$ for $x$ into $6 {(2)}^{x}$ .

$a = 6 \cdot 2^{1}$

Simplify.

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Evaluate the exponent.

$a = 6 \cdot 2$

Multiply $6$ by $2$ .

$a = 12$

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

$12 \frac{1 - 2^{10}}{1 - 1 \cdot 2}$

Simplify.

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Simplify the numerator.

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Raise $2$ to the power of $10$ .

$12 \frac{1 - 1 \cdot 1024}{1 - 1 \cdot 2}$

Multiply $- 1$ by $1024$ .

$12 \frac{1 - 1024}{1 - 1 \cdot 2}$

Subtract $1024$ from $1$ .

$12 \frac{- 1023}{1 - 1 \cdot 2}$

Simplify the denominator.

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Multiply $- 1$ by $2$ .

$12 \frac{- 1023}{1 - 2}$

Subtract $2$ from $1$ .

$12 (\frac{- 1023}{- 1})$

Divide $- 1023$ by $- 1$ .

$12 \cdot 1023$

Multiply $12$ by $1023$ .

$12276$