Precalculus Examples

$1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$

Step 1

This is a geometric sequence since there is a common ratio between each term. In this case, multiplying the previous term in the sequence by $\frac{1}{2}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{1}{2}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 1$ and $r = \frac{1}{2}$ .

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

Step 4

Multiply ${(\frac{1}{2})}^{n - 1}$ by $1$ .

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

Step 5

Apply the product rule to $\frac{1}{2}$ .

$a_{n} = \frac{1^{n - 1}}{2^{n - 1}}$

Step 6

One to any power is one.

$a_{n} = \frac{1}{2^{n - 1}}$

Step 7

This is the formula to find the sum of the first $n$ terms of the geometric sequence. To evaluate it, find the values of $r$ and $a_{1}$ .

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

Step 8

Replace the variables with the known values to find $S_{7}$ .

$S_{7} = 1 \cdot \frac{{(\frac{1}{2})}^{7} - 1}{\frac{1}{2} - 1}$

Step 9

Multiply $\frac{{(\frac{1}{2})}^{7} - 1}{\frac{1}{2} - 1}$ by $1$ .

$S_{7} = \frac{{(\frac{1}{2})}^{7} - 1}{\frac{1}{2} - 1}$

Step 10

Multiply the numerator and denominator of the fraction by $2$ .

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Step 10.1

Multiply $\frac{{(\frac{1}{2})}^{7} - 1}{\frac{1}{2} - 1}$ by $\frac{2}{2}$ .

$S_{7} = \frac{2}{2} \cdot \frac{{(\frac{1}{2})}^{7} - 1}{\frac{1}{2} - 1}$

Step 10.2

Combine.

$S_{7} = \frac{2 ({(\frac{1}{2})}^{7} - 1)}{2 (\frac{1}{2} - 1)}$

Step 11

Apply the distributive property.

$S_{7} = \frac{2 {(\frac{1}{2})}^{7} + 2 \cdot - 1}{2 (\frac{1}{2}) + 2 \cdot - 1}$

Step 12

Cancel the common factor of $2$ .

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Step 12.1

Cancel the common factor.

$S_{7} = \frac{2 {(\frac{1}{2})}^{7} + 2 \cdot - 1}{2 (\frac{1}{2}) + 2 \cdot - 1}$

Step 12.2

Rewrite the expression.

$S_{7} = \frac{2 {(\frac{1}{2})}^{7} + 2 \cdot - 1}{1 + 2 \cdot - 1}$

Step 13

Simplify the numerator.

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Step 13.1

Apply the product rule to $\frac{1}{2}$ .

$S_{7} = \frac{2 (\frac{1^{7}}{2^{7}}) + 2 \cdot - 1}{1 + 2 \cdot - 1}$

Step 13.2

Cancel the common factor of $2$ .

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Step 13.2.1

Factor $2$ out of $2^{7}$ .

$S_{7} = \frac{2 (\frac{1^{7}}{2 \cdot 2^{6}}) + 2 \cdot - 1}{1 + 2 \cdot - 1}$

Step 13.2.2

Cancel the common factor.

$S_{7} = \frac{2 (\frac{1^{7}}{2 \cdot 2^{6}}) + 2 \cdot - 1}{1 + 2 \cdot - 1}$

Step 13.2.3

Rewrite the expression.

$S_{7} = \frac{\frac{1^{7}}{2^{6}} + 2 \cdot - 1}{1 + 2 \cdot - 1}$

Step 13.3

One to any power is one.

$S_{7} = \frac{\frac{1}{2^{6}} + 2 \cdot - 1}{1 + 2 \cdot - 1}$

Step 13.4

Raise $2$ to the power of $6$ .

$S_{7} = \frac{\frac{1}{64} + 2 \cdot - 1}{1 + 2 \cdot - 1}$

Step 13.5

Multiply $2$ by $- 1$ .

$S_{7} = \frac{\frac{1}{64} - 2}{1 + 2 \cdot - 1}$

Step 13.6

To write $- 2$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$S_{7} = \frac{\frac{1}{64} - 2 \cdot \frac{64}{64}}{1 + 2 \cdot - 1}$

Step 13.7

Combine $- 2$ and $\frac{64}{64}$ .

$S_{7} = \frac{\frac{1}{64} + \frac{- 2 \cdot 64}{64}}{1 + 2 \cdot - 1}$

Step 13.8

Combine the numerators over the common denominator.

$S_{7} = \frac{\frac{1 - 2 \cdot 64}{64}}{1 + 2 \cdot - 1}$

Step 13.9

Simplify the numerator.

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Step 13.9.1

Multiply $- 2$ by $64$ .

$S_{7} = \frac{\frac{1 - 128}{64}}{1 + 2 \cdot - 1}$

Step 13.9.2

Subtract $128$ from $1$ .

$S_{7} = \frac{\frac{- 127}{64}}{1 + 2 \cdot - 1}$

Step 13.10

Move the negative in front of the fraction.

$S_{7} = \frac{- \frac{127}{64}}{1 + 2 \cdot - 1}$

Step 14

Simplify the denominator.

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Step 14.1

Multiply $2$ by $- 1$ .

$S_{7} = \frac{- \frac{127}{64}}{1 - 2}$

Step 14.2

Subtract $2$ from $1$ .

$S_{7} = \frac{- \frac{127}{64}}{- 1}$

Step 15

Reduce the expression by cancelling the common factors.

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Step 15.1

Dividing two negative values results in a positive value.

$S_{7} = \frac{\frac{127}{64}}{1}$

Step 15.2

Divide $\frac{127}{64}$ by $1$ .

$S_{7} = \frac{127}{64}$