Calculus Examples

$\sum_{i = 1}^{100} 2^{i}$

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 $r$ .

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

Step 2.2

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

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factors.

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Step 2.2.2.1

Multiply by $1$ .

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

Step 2.2.2.2

Cancel the common factor.

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

Step 2.2.2.3

Rewrite the expression.

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

Step 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 $2^{i}$ .

$a = 2^{1}$

Step 3.2

Evaluate the exponent.

$a = 2$

Step 4

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

$2 \frac{1 - 2^{100}}{1 - 1 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Multiply $- 1$ by $2$ .

$2 \frac{1 - 2^{100}}{1 - 2}$

Step 5.1.2

Subtract $2$ from $1$ .

$2 \frac{1 - 2^{100}}{- 1}$

Step 5.2

Move the negative one from the denominator of $\frac{1 - 2^{100}}{- 1}$ .

$2 (- 1 \cdot (1 - 2^{100}))$

Step 5.3

Rewrite $- 1 \cdot (1 - 2^{100})$ as $- (1 - 2^{100})$ .

$2 (- (1 - 2^{100}))$

Step 5.4

Apply the distributive property.

$2 (- 1 \cdot 1 - - 2^{100})$

Step 5.5

Multiply $- 1$ by $1$ .

$2 (- 1 - - 2^{100})$

Step 5.6

Multiply $- - 2^{100}$ .

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Step 5.6.1

Multiply $- 1$ by $- 1$ .

$2 (- 1 + 1 \cdot 2^{100})$

Step 5.6.2

Multiply $2^{100}$ by $1$ .

$2 (- 1 + 2^{100})$

Step 5.7

Apply the distributive property.

$2 \cdot - 1 + 2 \cdot 2^{100}$

Step 5.8

Multiply $2$ by $- 1$ .

$- 2 + 2 \cdot 2^{100}$

Step 5.9

Multiply $2$ by $2^{100}$ by adding the exponents.

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Step 5.9.1

Multiply $2$ by $2^{100}$ .

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Step 5.9.1.1

Raise $2$ to the power of $1$ .

$- 2 + 2^{1} \cdot 2^{100}$

Step 5.9.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- 2 + 2^{1 + 100}$

Step 5.9.2

Add $1$ and $100$ .

$- 2 + 2^{101}$