Calculus Examples

$\sum_{n = 1}^{5} 2^{n - 2}$

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

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

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

Simplify.

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

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

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

Cancel the common factors.

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Multiply by $1$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Divide $2^{n - 1 - (n - 2)}$ by $1$ .

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

Simplify each term.

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Apply the distributive property.

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

Multiply $- 1$ by $- 2$ .

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

Combine the opposite terms in $n - 1 - n + 2$ .

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Subtract $n$ from $n$ .

$r = 2^{0 - 1 + 2}$

Subtract $1$ from $0$ .

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

Add $- 1$ and $2$ .

$r = 2^{1}$

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

$a = 2^{1 - 2}$

Simplify.

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Subtract $2$ from $1$ .

$a = 2^{- 1}$

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 4

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

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

Step 5

Simplify.

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Simplify the numerator.

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Raise $2$ to the power of $5$ .

$\frac{1}{2} \cdot \frac{1 - 1 \cdot 32}{1 - 1 \cdot 2}$

Multiply $- 1$ by $32$ .

$\frac{1}{2} \cdot \frac{1 - 32}{1 - 1 \cdot 2}$

Subtract $32$ from $1$ .

$\frac{1}{2} \cdot \frac{- 31}{1 - 1 \cdot 2}$

Simplify the denominator.

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Multiply $- 1$ by $2$ .

$\frac{1}{2} \cdot \frac{- 31}{1 - 2}$

Subtract $2$ from $1$ .

$\frac{1}{2} \cdot \frac{- 31}{- 1}$

Divide $- 31$ by $- 1$ .

$\frac{1}{2} \cdot 31$

Combine $\frac{1}{2}$ and $31$ .

$\frac{31}{2}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$\frac{31}{2}$

Decimal Form:

$15.5$

Mixed Number Form:

$15 \frac{1}{2}$