Precalculus Examples

$\sum_{i = 1}^{50} 2^{2 i}$

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 $r$ .

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

Step 2.2

Simplify.

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Step 2.2.1

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

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Step 2.2.1.1

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

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

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Multiply by $1$ .

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

Step 2.2.1.2.2

Cancel the common factor.

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

Step 2.2.1.2.3

Rewrite the expression.

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

Step 2.2.1.2.4

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

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

Step 2.2.2

Simplify each term.

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Step 2.2.2.1

Apply the distributive property.

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

Step 2.2.2.2

Multiply $2$ by $1$ .

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

Step 2.2.3

Combine the opposite terms in $2 i + 2 - 2 i$ .

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Step 2.2.3.1

Subtract $2 i$ from $2 i$ .

$r = 2^{0 + 2}$

Step 2.2.3.2

Add $0$ and $2$ .

$r = 2^{2}$

Step 2.2.4

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

$r = 4$

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 $2^{2 i}$ .

$a = 2^{2 \cdot 1}$

Step 3.2

Simplify.

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Step 3.2.1

Multiply $2$ by $1$ .

$a = 2^{2}$

Step 3.2.2

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

$a = 4$

Step 4

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

$4 \frac{1 - 4^{50}}{1 - 1 \cdot 4}$

Step 5

Simplify.

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Step 5.1

Simplify the denominator.

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Step 5.1.1

Multiply $- 1$ by $4$ .

$4 \frac{1 - 4^{50}}{1 - 4}$

Step 5.1.2

Subtract $4$ from $1$ .

$4 \frac{1 - 4^{50}}{- 3}$

Step 5.2

Move the negative in front of the fraction.

$4 (- \frac{1 - 4^{50}}{3})$

Step 5.3

Multiply $4 (- \frac{1 - 4^{50}}{3})$ .

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Step 5.3.1

Multiply $- 1$ by $4$ .

$- 4 \frac{1 - 4^{50}}{3}$

Step 5.3.2

Combine $- 4$ and $\frac{1 - 4^{50}}{3}$ .

$\frac{- 4 (1 - 4^{50})}{3}$

Step 5.4

Move the negative in front of the fraction.

$- \frac{4 (1 - 4^{50})}{3}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$- \frac{4 (1 - 4^{50})}{3}$

Decimal Form:

$1.6902008 \cdot 10^{30}$