Calculus Examples

$3$ , $\frac{3}{4}$ , $\frac{3}{16}$ , $\frac{3}{64}$

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 $a_{1} = 3$ and $r = \frac{1}{4}$ .

$a_{n} = 3 {(\frac{1}{4})}^{n - 1}$

Step 4

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

$a_{n} = 3 \frac{1^{n - 1}}{4^{n - 1}}$

Step 5

One to any power is one.

$a_{n} = 3 \frac{1}{4^{n - 1}}$

Step 6

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

$a_{n} = \frac{3}{4^{n - 1}}$

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 $S_{4}$ .

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

Step 9

Simplify the numerator.

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Step 9.1

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

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

Step 9.2

One to any power is one.

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

Step 9.3

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

$S_{4} = 3 \cdot \frac{\frac{1}{256} - 1}{\frac{1}{4} - 1}$

Step 9.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{256}{256}$ .

$S_{4} = 3 \cdot \frac{\frac{1}{256} - 1 \cdot \frac{256}{256}}{\frac{1}{4} - 1}$

Step 9.5

Combine $- 1$ and $\frac{256}{256}$ .

$S_{4} = 3 \cdot \frac{\frac{1}{256} + \frac{- 1 \cdot 256}{256}}{\frac{1}{4} - 1}$

Step 9.6

Combine the numerators over the common denominator.

$S_{4} = 3 \cdot \frac{\frac{1 - 1 \cdot 256}{256}}{\frac{1}{4} - 1}$

Step 9.7

Simplify the numerator.

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Step 9.7.1

Multiply $- 1$ by $256$ .

$S_{4} = 3 \cdot \frac{\frac{1 - 256}{256}}{\frac{1}{4} - 1}$

Step 9.7.2

Subtract $256$ from $1$ .

$S_{4} = 3 \cdot \frac{\frac{- 255}{256}}{\frac{1}{4} - 1}$

Step 9.8

Move the negative in front of the fraction.

$S_{4} = 3 \cdot \frac{- \frac{255}{256}}{\frac{1}{4} - 1}$

Step 10

Simplify the denominator.

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Step 10.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$S_{4} = 3 \cdot \frac{- \frac{255}{256}}{\frac{1}{4} - 1 \cdot \frac{4}{4}}$

Step 10.2

Combine $- 1$ and $\frac{4}{4}$ .

$S_{4} = 3 \cdot \frac{- \frac{255}{256}}{\frac{1}{4} + \frac{- 1 \cdot 4}{4}}$

Step 10.3

Combine the numerators over the common denominator.

$S_{4} = 3 \cdot \frac{- \frac{255}{256}}{\frac{1 - 1 \cdot 4}{4}}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Multiply $- 1$ by $4$ .

$S_{4} = 3 \cdot \frac{- \frac{255}{256}}{\frac{1 - 4}{4}}$

Step 10.4.2

Subtract $4$ from $1$ .

$S_{4} = 3 \cdot \frac{- \frac{255}{256}}{\frac{- 3}{4}}$

Step 10.5

Move the negative in front of the fraction.

$S_{4} = 3 \cdot \frac{- \frac{255}{256}}{- \frac{3}{4}}$

Step 11

Dividing two negative values results in a positive value.

$S_{4} = 3 \cdot \frac{\frac{255}{256}}{\frac{3}{4}}$

Step 12

Multiply the numerator by the reciprocal of the denominator.

$S_{4} = 3 \cdot (\frac{255}{256} \cdot \frac{4}{3})$

Step 13

Cancel the common factor of $3$ .

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Step 13.1

Factor $3$ out of $255$ .

$S_{4} = 3 \cdot (\frac{3 (85)}{256} \cdot \frac{4}{3})$

Step 13.2

Cancel the common factor.

$S_{4} = 3 \cdot (\frac{3 \cdot 85}{256} \cdot \frac{4}{3})$

Step 13.3

Rewrite the expression.

$S_{4} = 3 \cdot (\frac{85}{256} \cdot 4)$

Step 14

Cancel the common factor of $4$ .

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Step 14.1

Factor $4$ out of $256$ .

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

Step 14.2

Cancel the common factor.

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

Step 14.3

Rewrite the expression.

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

Step 15

Multiply $3 (\frac{85}{64})$ .

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Step 15.1

Combine $3$ and $\frac{85}{64}$ .

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

Step 15.2

Multiply $3$ by $85$ .

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