Calculus Examples

$\sum_{j = 1}^{3} 10 {(\frac{1}{2})}^{j}$

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

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

$r = \frac{10 {(\frac{1}{2})}^{j + 1}}{10 {(\frac{1}{2})}^{j}}$

Simplify.

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Cancel the common factor of $10$ .

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Cancel the common factor.

$r = \frac{10 {(\frac{1}{2})}^{j + 1}}{10 {(\frac{1}{2})}^{j}}$

Rewrite the expression.

$r = \frac{{(\frac{1}{2})}^{j + 1}}{{(\frac{1}{2})}^{j}}$

Cancel the common factor of ${(\frac{1}{2})}^{j + 1}$ and ${(\frac{1}{2})}^{j}$ .

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Factor ${(\frac{1}{2})}^{j}$ out of ${(\frac{1}{2})}^{j + 1}$ .

$r = \frac{{(\frac{1}{2})}^{j} \frac{1}{2}}{{(\frac{1}{2})}^{j}}$

Cancel the common factors.

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Multiply by $1$ .

$r = \frac{{(\frac{1}{2})}^{j} \frac{1}{2}}{{(\frac{1}{2})}^{j} \cdot 1}$

Cancel the common factor.

$r = \frac{{(\frac{1}{2})}^{j} \frac{1}{2}}{{(\frac{1}{2})}^{j} \cdot 1}$

Rewrite the expression.

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

Divide $\frac{1}{2}$ by $1$ .

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

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

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Substitute $1$ for $j$ into $10 {(\frac{1}{2})}^{j}$ .

$a = 10 {(\frac{1}{2})}^{1}$

Simplify.

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Simplify.

$a = 10 (\frac{1}{2})$

Cancel the common factor of $2$ .

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Factor $2$ out of $10$ .

$a = 2 (5) \frac{1}{2}$

Cancel the common factor.

$a = 2 \cdot 5 \frac{1}{2}$

Rewrite the expression.

$a = 5$

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

$5 \frac{1 - {(\frac{1}{2})}^{3}}{1 - \frac{1}{2}}$

Simplify.

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Multiply the numerator and denominator of the complex fraction by $2$ .

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Multiply $\frac{1 - {(\frac{1}{2})}^{3}}{1 - \frac{1}{2}}$ by $\frac{2}{2}$ .

$5 (\frac{2}{2} \cdot \frac{1 - {(\frac{1}{2})}^{3}}{1 - \frac{1}{2}})$

Combine.

$5 \frac{2 (1 - {(\frac{1}{2})}^{3})}{2 (1 - \frac{1}{2})}$

Apply the distributive property.

$5 \frac{2 \cdot 1 + 2 (- {(\frac{1}{2})}^{3})}{2 \cdot 1 + 2 (- \frac{1}{2})}$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{1}{2}$ into the numerator.

$5 \frac{2 \cdot 1 + 2 (- {(\frac{1}{2})}^{3})}{2 \cdot 1 + 2 (\frac{- 1}{2})}$

Cancel the common factor.

$5 \frac{2 \cdot 1 + 2 (- {(\frac{1}{2})}^{3})}{2 \cdot 1 + 2 (\frac{- 1}{2})}$

Rewrite the expression.

$5 \frac{2 \cdot 1 + 2 (- {(\frac{1}{2})}^{3})}{2 \cdot 1 - 1}$

Simplify the numerator.

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Multiply $2$ by $1$ .

$5 \frac{2 + 2 (- {(\frac{1}{2})}^{3})}{2 \cdot 1 - 1}$

Apply the product rule to $\frac{1}{2}$ .

$5 \frac{2 + 2 (- \frac{1^{3}}{2^{3}})}{2 \cdot 1 - 1}$

Cancel the common factor of $2$ .

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Move the leading negative in $- \frac{1^{3}}{2^{3}}$ into the numerator.

$5 \frac{2 + 2 \frac{- 1^{3}}{2^{3}}}{2 \cdot 1 - 1}$

Factor $2$ out of $2^{3}$ .

$5 \frac{2 + 2 \frac{- 1^{3}}{2 \cdot 2^{2}}}{2 \cdot 1 - 1}$

Cancel the common factor.

$5 \frac{2 + 2 \frac{- 1^{3}}{2 \cdot 2^{2}}}{2 \cdot 1 - 1}$

Rewrite the expression.

$5 \frac{2 + \frac{- 1^{3}}{2^{2}}}{2 \cdot 1 - 1}$

One to any power is one.

$5 \frac{2 + \frac{- 1 \cdot 1}{2^{2}}}{2 \cdot 1 - 1}$

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

$5 \frac{2 + \frac{- 1 \cdot 1}{4}}{2 \cdot 1 - 1}$

Multiply $- 1$ by $1$ .

$5 \frac{2 + \frac{- 1}{4}}{2 \cdot 1 - 1}$

Move the negative in front of the fraction.

$5 \frac{2 - \frac{1}{4}}{2 \cdot 1 - 1}$

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$5 \frac{2 \cdot \frac{4}{4} - \frac{1}{4}}{2 \cdot 1 - 1}$

Combine $2$ and $\frac{4}{4}$ .

$5 \frac{\frac{2 \cdot 4}{4} - \frac{1}{4}}{2 \cdot 1 - 1}$

Combine the numerators over the common denominator.

$5 \frac{\frac{2 \cdot 4 - 1}{4}}{2 \cdot 1 - 1}$

Simplify the numerator.

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Multiply $2$ by $4$ .

$5 \frac{\frac{8 - 1}{4}}{2 \cdot 1 - 1}$

Subtract $1$ from $8$ .

$5 \frac{\frac{7}{4}}{2 \cdot 1 - 1}$

Simplify the denominator.

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Multiply $2$ by $1$ .

$5 \frac{\frac{7}{4}}{2 - 1}$

Subtract $1$ from $2$ .

$5 \frac{\frac{7}{4}}{1}$

Divide $\frac{7}{4}$ by $1$ .

$5 (\frac{7}{4})$

Multiply $5 (\frac{7}{4})$ .

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Combine $5$ and $\frac{7}{4}$ .

$\frac{5 \cdot 7}{4}$

Multiply $5$ by $7$ .

$\frac{35}{4}$

The result can be shown in multiple forms.

Exact Form:

$\frac{35}{4}$

Decimal Form:

$8.75$

Mixed Number Form:

$8 \frac{3}{4}$