Precalculus Examples

$1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$

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 $\frac{1}{4}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{1}{4}$

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 $r = \frac{1}{4}$ .

$a_{n} = 1 {(\frac{1}{4})}^{n - 1}$

Step 4

Multiply ${(\frac{1}{4})}^{n - 1}$ by $1$ .

$a_{n} = {(\frac{1}{4})}^{n - 1}$

Step 5

Apply the product rule to $\frac{1}{4}$ .

$a_{n} = \frac{1^{n - 1}}{4^{n - 1}}$

Step 6

One to any power is one.

$a_{n} = \frac{1}{4^{n - 1}}$

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 $S_{4}$ .

$S_{4} = 1 \cdot \frac{{(\frac{1}{4})}^{4} - 1}{\frac{1}{4} - 1}$

Step 9

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

$S_{4} = \frac{{(\frac{1}{4})}^{4} - 1}{\frac{1}{4} - 1}$

Step 10

Multiply the numerator and denominator of the fraction by $4$ .

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Step 10.1

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

$S_{4} = \frac{4}{4} \cdot \frac{{(\frac{1}{4})}^{4} - 1}{\frac{1}{4} - 1}$

Step 10.2

Combine.

$S_{4} = \frac{4 ({(\frac{1}{4})}^{4} - 1)}{4 (\frac{1}{4} - 1)}$

Step 11

Apply the distributive property.

$S_{4} = \frac{4 {(\frac{1}{4})}^{4} + 4 \cdot - 1}{4 (\frac{1}{4}) + 4 \cdot - 1}$

Step 12

Cancel the common factor of $4$ .

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Step 12.1

Cancel the common factor.

$S_{4} = \frac{4 {(\frac{1}{4})}^{4} + 4 \cdot - 1}{4 (\frac{1}{4}) + 4 \cdot - 1}$

Step 12.2

Rewrite the expression.

$S_{4} = \frac{4 {(\frac{1}{4})}^{4} + 4 \cdot - 1}{1 + 4 \cdot - 1}$

Step 13

Simplify the numerator.

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Step 13.1

Apply the product rule to $\frac{1}{4}$ .

$S_{4} = \frac{4 (\frac{1^{4}}{4^{4}}) + 4 \cdot - 1}{1 + 4 \cdot - 1}$

Step 13.2

Cancel the common factor of $4$ .

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Step 13.2.1

Factor $4$ out of $4^{4}$ .

$S_{4} = \frac{4 (\frac{1^{4}}{4 \cdot 4^{3}}) + 4 \cdot - 1}{1 + 4 \cdot - 1}$

Step 13.2.2

Cancel the common factor.

$S_{4} = \frac{4 (\frac{1^{4}}{4 \cdot 4^{3}}) + 4 \cdot - 1}{1 + 4 \cdot - 1}$

Step 13.2.3

Rewrite the expression.

$S_{4} = \frac{\frac{1^{4}}{4^{3}} + 4 \cdot - 1}{1 + 4 \cdot - 1}$

Step 13.3

One to any power is one.

$S_{4} = \frac{\frac{1}{4^{3}} + 4 \cdot - 1}{1 + 4 \cdot - 1}$

Step 13.4

Raise $4$ to the power of $3$ .

$S_{4} = \frac{\frac{1}{64} + 4 \cdot - 1}{1 + 4 \cdot - 1}$

Step 13.5

Multiply $4$ by $- 1$ .

$S_{4} = \frac{\frac{1}{64} - 4}{1 + 4 \cdot - 1}$

Step 13.6

To write $- 4$ as a fraction with a common denominator, multiply by $\frac{64}{64}$ .

$S_{4} = \frac{\frac{1}{64} - 4 \cdot \frac{64}{64}}{1 + 4 \cdot - 1}$

Step 13.7

Combine $- 4$ and $\frac{64}{64}$ .

$S_{4} = \frac{\frac{1}{64} + \frac{- 4 \cdot 64}{64}}{1 + 4 \cdot - 1}$

Step 13.8

Combine the numerators over the common denominator.

$S_{4} = \frac{\frac{1 - 4 \cdot 64}{64}}{1 + 4 \cdot - 1}$

Step 13.9

Simplify the numerator.

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Step 13.9.1

Multiply $- 4$ by $64$ .

$S_{4} = \frac{\frac{1 - 256}{64}}{1 + 4 \cdot - 1}$

Step 13.9.2

Subtract $256$ from $1$ .

$S_{4} = \frac{\frac{- 255}{64}}{1 + 4 \cdot - 1}$

Step 13.10

Move the negative in front of the fraction.

$S_{4} = \frac{- \frac{255}{64}}{1 + 4 \cdot - 1}$

Step 14

Simplify the denominator.

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Step 14.1

Multiply $4$ by $- 1$ .

$S_{4} = \frac{- \frac{255}{64}}{1 - 4}$

Step 14.2

Subtract $4$ from $1$ .

$S_{4} = \frac{- \frac{255}{64}}{- 3}$

Step 15

Multiply the numerator by the reciprocal of the denominator.

$S_{4} = - \frac{255}{64} \cdot \frac{1}{- 3}$

Step 16

Cancel the common factor of $3$ .

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Step 16.1

Move the leading negative in $- \frac{255}{64}$ into the numerator.

$S_{4} = \frac{- 255}{64} \cdot \frac{1}{- 3}$

Step 16.2

Factor $3$ out of $- 255$ .

$S_{4} = \frac{3 (- 85)}{64} \cdot \frac{1}{- 3}$

Step 16.3

Factor $3$ out of $- 3$ .

$S_{4} = \frac{3 \cdot - 85}{64} \cdot \frac{1}{3 \cdot - 1}$

Step 16.4

Cancel the common factor.

$S_{4} = \frac{3 \cdot - 85}{64} \cdot \frac{1}{3 \cdot - 1}$

Step 16.5

Rewrite the expression.

$S_{4} = \frac{- 85}{64} \cdot \frac{1}{- 1}$

Step 17

Multiply $\frac{- 85}{64}$ by $\frac{1}{- 1}$ .

$S_{4} = \frac{- 85}{64 \cdot - 1}$

Step 18

Multiply $64$ by $- 1$ .

$S_{4} = \frac{- 85}{- 64}$

Step 19

Dividing two negative values results in a positive value.

$S_{4} = \frac{85}{64}$