Calculus Examples

$\sum_{i = 1}^{25} 2^{i} - 2^{i - 1}$

Step 1

Split the summation into smaller summations that fit the summation rules.

$\sum_{k = 1}^{25} 2^{i} - 2^{i - 1} = \sum_{k = 1}^{25} 2^{i} + \sum_{k = 1}^{25} - 2^{i - 1}$

Step 2

Evaluate $\sum_{k = 1}^{25} 2^{i}$ .

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Step 2.1

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 2.2

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

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Step 2.2.1

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

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

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 2.3

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

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Step 2.3.1

Substitute $1$ for $i$ into $2^{i}$ .

$a = 2^{1}$

Step 2.3.2

Evaluate the exponent.

$a = 2$

Step 2.4

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

$2 \frac{1 - 2^{25}}{1 - 1 \cdot 2}$

Step 2.5

Simplify.

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Step 2.5.1

Simplify the numerator.

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Step 2.5.1.1

Raise $2$ to the power of $25$ .

$2 \frac{1 - 1 \cdot 33554432}{1 - 1 \cdot 2}$

Step 2.5.1.2

Multiply $- 1$ by $33554432$ .

$2 \frac{1 - 33554432}{1 - 1 \cdot 2}$

Step 2.5.1.3

Subtract $33554432$ from $1$ .

$2 \frac{- 33554431}{1 - 1 \cdot 2}$

Step 2.5.2

Simplify the denominator.

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Step 2.5.2.1

Multiply $- 1$ by $2$ .

$2 \frac{- 33554431}{1 - 2}$

Step 2.5.2.2

Subtract $2$ from $1$ .

$2 (\frac{- 33554431}{- 1})$

Step 2.5.3

Divide $- 33554431$ by $- 1$ .

$2 \cdot 33554431$

Step 2.5.4

Multiply $2$ by $33554431$ .

$67108862$

Step 3

Evaluate $\sum_{k = 1}^{25} - 2^{i - 1}$ .

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Step 3.1

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 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Simplify.

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Step 3.2.2.1

Dividing two negative values results in a positive value.

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

Step 3.2.2.2

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

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Step 3.2.2.2.1

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

$r = \frac{2^{i - 1} \cdot 2^{i + 0 - (i - 1)}}{2^{i - 1}}$

Step 3.2.2.2.2

Cancel the common factors.

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Step 3.2.2.2.2.1

Multiply by $1$ .

$r = \frac{2^{i - 1} \cdot 2^{i + 0 - (i - 1)}}{2^{i - 1} \cdot 1}$

Step 3.2.2.2.2.2

Cancel the common factor.

$r = \frac{2^{i - 1} \cdot 2^{i + 0 - (i - 1)}}{2^{i - 1} \cdot 1}$

Step 3.2.2.2.2.3

Rewrite the expression.

$r = \frac{2^{i + 0 - (i - 1)}}{1}$

Step 3.2.2.2.2.4

Divide $2^{i + 0 - (i - 1)}$ by $1$ .

$r = 2^{i + 0 - (i - 1)}$

Step 3.2.2.3

Add $i$ and $0$ .

$r = 2^{i - (i - 1)}$

Step 3.2.2.4

Simplify each term.

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Step 3.2.2.4.1

Apply the distributive property.

$r = 2^{i - i - - 1}$

Step 3.2.2.4.2

Multiply $- 1$ by $- 1$ .

$r = 2^{i - i + 1}$

Step 3.2.2.5

Subtract $i$ from $i$ .

$r = 2^{0 + 1}$

Step 3.2.2.6

Add $0$ and $1$ .

$r = 2^{1}$

Step 3.2.2.7

Evaluate the exponent.

$r = 2$

Step 3.3

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

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Step 3.3.1

Substitute $1$ for $i$ into $- 2^{i - 1}$ .

$a = - 2^{1 - 1}$

Step 3.3.2

Simplify.

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Step 3.3.2.1

Subtract $1$ from $1$ .

$a = - 2^{0}$

Step 3.3.2.2

Anything raised to $0$ is $1$ .

$a = - 1 \cdot 1$

Step 3.3.2.3

Multiply $- 1$ by $1$ .

$a = - 1$

Step 3.4

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

$- \frac{1 - 2^{25}}{1 - 1 \cdot 2}$

Step 3.5

Simplify.

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Step 3.5.1

Simplify the numerator.

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Step 3.5.1.1

Raise $2$ to the power of $25$ .

$- \frac{1 - 1 \cdot 33554432}{1 - 1 \cdot 2}$

Step 3.5.1.2

Multiply $- 1$ by $33554432$ .

$- \frac{1 - 33554432}{1 - 1 \cdot 2}$

Step 3.5.1.3

Subtract $33554432$ from $1$ .

$- \frac{- 33554431}{1 - 1 \cdot 2}$

Step 3.5.2

Simplify the denominator.

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Step 3.5.2.1

Multiply $- 1$ by $2$ .

$- \frac{- 33554431}{1 - 2}$

Step 3.5.2.2

Subtract $2$ from $1$ .

$- \frac{- 33554431}{- 1}$

Step 3.5.3

Divide $- 33554431$ by $- 1$ .

$- 1 \cdot 33554431$

Step 3.5.4

Multiply $- 1$ by $33554431$ .

$- 33554431$

Step 4

Add the results of the summations.

$67108862 - 33554431$

Step 5

Subtract $33554431$ from $67108862$ .

$33554431$