Calculus Examples

$135$ , $45$ , $15$

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 $\frac{1}{3}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{1}{3}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 135$ and $r = \frac{1}{3}$ .

$a_{n} = 135 {(\frac{1}{3})}^{n - 1}$

Step 4

Apply the product rule to $\frac{1}{3}$ .

$a_{n} = 135 \frac{1^{n - 1}}{3^{n - 1}}$

Step 5

One to any power is one.

$a_{n} = 135 \frac{1}{3^{n - 1}}$

Step 6

Combine $135$ and $\frac{1}{3^{n - 1}}$ .

$a_{n} = \frac{135}{3^{n - 1}}$

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 $S_{5}$ .

$S_{5} = 135 \cdot \frac{{(\frac{1}{3})}^{5} - 1}{\frac{1}{3} - 1}$

Step 9

Simplify the numerator.

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Step 9.1

Apply the product rule to $\frac{1}{3}$ .

$S_{5} = 135 \cdot \frac{\frac{1^{5}}{3^{5}} - 1}{\frac{1}{3} - 1}$

Step 9.2

One to any power is one.

$S_{5} = 135 \cdot \frac{\frac{1}{3^{5}} - 1}{\frac{1}{3} - 1}$

Step 9.3

Raise $3$ to the power of $5$ .

$S_{5} = 135 \cdot \frac{\frac{1}{243} - 1}{\frac{1}{3} - 1}$

Step 9.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{243}{243}$ .

$S_{5} = 135 \cdot \frac{\frac{1}{243} - 1 \cdot \frac{243}{243}}{\frac{1}{3} - 1}$

Step 9.5

Combine $- 1$ and $\frac{243}{243}$ .

$S_{5} = 135 \cdot \frac{\frac{1}{243} + \frac{- 1 \cdot 243}{243}}{\frac{1}{3} - 1}$

Step 9.6

Combine the numerators over the common denominator.

$S_{5} = 135 \cdot \frac{\frac{1 - 1 \cdot 243}{243}}{\frac{1}{3} - 1}$

Step 9.7

Simplify the numerator.

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Step 9.7.1

Multiply $- 1$ by $243$ .

$S_{5} = 135 \cdot \frac{\frac{1 - 243}{243}}{\frac{1}{3} - 1}$

Step 9.7.2

Subtract $243$ from $1$ .

$S_{5} = 135 \cdot \frac{\frac{- 242}{243}}{\frac{1}{3} - 1}$

Step 9.8

Move the negative in front of the fraction.

$S_{5} = 135 \cdot \frac{- \frac{242}{243}}{\frac{1}{3} - 1}$

Step 10

Simplify the denominator.

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Step 10.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$S_{5} = 135 \cdot \frac{- \frac{242}{243}}{\frac{1}{3} - 1 \cdot \frac{3}{3}}$

Step 10.2

Combine $- 1$ and $\frac{3}{3}$ .

$S_{5} = 135 \cdot \frac{- \frac{242}{243}}{\frac{1}{3} + \frac{- 1 \cdot 3}{3}}$

Step 10.3

Combine the numerators over the common denominator.

$S_{5} = 135 \cdot \frac{- \frac{242}{243}}{\frac{1 - 1 \cdot 3}{3}}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Multiply $- 1$ by $3$ .

$S_{5} = 135 \cdot \frac{- \frac{242}{243}}{\frac{1 - 3}{3}}$

Step 10.4.2

Subtract $3$ from $1$ .

$S_{5} = 135 \cdot \frac{- \frac{242}{243}}{\frac{- 2}{3}}$

Step 10.5

Move the negative in front of the fraction.

$S_{5} = 135 \cdot \frac{- \frac{242}{243}}{- \frac{2}{3}}$

Step 11

Dividing two negative values results in a positive value.

$S_{5} = 135 \cdot \frac{\frac{242}{243}}{\frac{2}{3}}$

Step 12

Multiply the numerator by the reciprocal of the denominator.

$S_{5} = 135 \cdot (\frac{242}{243} \cdot \frac{3}{2})$

Step 13

Cancel the common factor of $2$ .

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Step 13.1

Factor $2$ out of $242$ .

$S_{5} = 135 \cdot (\frac{2 (121)}{243} \cdot \frac{3}{2})$

Step 13.2

Cancel the common factor.

$S_{5} = 135 \cdot (\frac{2 \cdot 121}{243} \cdot \frac{3}{2})$

Step 13.3

Rewrite the expression.

$S_{5} = 135 \cdot (\frac{121}{243} \cdot 3)$

Step 14

Cancel the common factor of $3$ .

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Step 14.1

Factor $3$ out of $243$ .

$S_{5} = 135 \cdot (\frac{121}{3 (81)} \cdot 3)$

Step 14.2

Cancel the common factor.

$S_{5} = 135 \cdot (\frac{121}{3 \cdot 81} \cdot 3)$

Step 14.3

Rewrite the expression.

$S_{5} = 135 \cdot \frac{121}{81}$

Step 15

Cancel the common factor of $27$ .

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Step 15.1

Factor $27$ out of $135$ .

$S_{5} = 27 (5) \cdot \frac{121}{81}$

Step 15.2

Factor $27$ out of $81$ .

$S_{5} = 27 \cdot 5 \cdot \frac{121}{27 \cdot 3}$

Step 15.3

Cancel the common factor.

$S_{5} = 27 \cdot 5 \cdot \frac{121}{27 \cdot 3}$

Step 15.4

Rewrite the expression.

$S_{5} = 5 \cdot \frac{121}{3}$

Step 16

Combine $5$ and $\frac{121}{3}$ .

$S_{5} = \frac{5 \cdot 121}{3}$

Step 17

Multiply $5$ by $121$ .

$S_{5} = \frac{605}{3}$

Step 18

Convert the fraction to a decimal.

$S_{5} = 201.\overline{6}$

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