Calculus Examples

$\sum_{i = 1}^{11} 2^{i - 1}$

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 $r$ .

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

Step 2.2

Simplify.

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Step 2.2.1

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

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Step 2.2.1.1

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 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Multiply by $1$ .

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

Step 2.2.1.2.2

Cancel the common factor.

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

Step 2.2.1.2.3

Rewrite the expression.

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

Step 2.2.1.2.4

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

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

Step 2.2.2

Add $i$ and $0$ .

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

Step 2.2.3

Simplify each term.

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Step 2.2.3.1

Apply the distributive property.

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

Step 2.2.3.2

Multiply $- 1$ by $- 1$ .

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

Step 2.2.4

Subtract $i$ from $i$ .

$r = 2^{0 + 1}$

Step 2.2.5

Add $0$ and $1$ .

$r = 2^{1}$

Step 2.2.6

Evaluate the exponent.

$r = 2$

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 $2^{i - 1}$ .

$a = 2^{1 - 1}$

Step 3.2

Simplify.

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Step 3.2.1

Subtract $1$ from $1$ .

$a = 2^{0}$

Step 3.2.2

Anything raised to $0$ is $1$ .

$a = 1$

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Multiply $\frac{1 - 2^{11}}{1 - 1 \cdot 2}$ by $1$ .

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

Step 5.2

Simplify the numerator.

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Step 5.2.1

Raise $2$ to the power of $11$ .

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

Step 5.2.2

Multiply $- 1$ by $2048$ .

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

Step 5.2.3

Subtract $2048$ from $1$ .

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

Step 5.3

Simplify the denominator.

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Step 5.3.1

Multiply $- 1$ by $2$ .

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

Step 5.3.2

Subtract $2$ from $1$ .

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

Step 5.4

Divide $- 2047$ by $- 1$ .

$2047$