Precalculus Examples

20 + 10 + 5 + \frac{5}{2}

$20 + 10 + 5 + \frac{5}{2}$

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}

$\frac{1}{2}$ gives the next term. In other words,

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

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

Geometric Sequence:

r = \frac{1}{2}

$r = \frac{1}{2}$

Step 2

This is the form of a geometric sequence.

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

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

Step 3

Substitute in the values of

a_{1} = 20

$a_{1} = 20$ and

r = \frac{1}{2}

$r = \frac{1}{2}$ .

a_{n} = 20 {(\frac{1}{2})}^{n - 1}

$a_{n} = 20 {(\frac{1}{2})}^{n - 1}$

Step 4

Apply the product rule to

\frac{1}{2}

$\frac{1}{2}$ .

a_{n} = 20 \frac{1^{n - 1}}{2^{n - 1}}

$a_{n} = 20 \frac{1^{n - 1}}{2^{n - 1}}$

Step 5

One to any power is one.

a_{n} = 20 \frac{1}{2^{n - 1}}

$a_{n} = 20 \frac{1}{2^{n - 1}}$

Step 6

Combine

20

$20$ and

\frac{1}{2^{n - 1}}

$\frac{1}{2^{n - 1}}$ .

a_{n} = \frac{20}{2^{n - 1}}

$a_{n} = \frac{20}{2^{n - 1}}$

Step 7

This is the formula to find the sum of the first

n

$n$ terms of the geometric sequence. To evaluate it, find the values of

r

$r$ and

a_{1}

$a_{1}$ .

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

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

Step 8

Replace the variables with the known values to find

S_{4}

$S_{4}$ .

S_{4} = 20 \cdot \frac{{(\frac{1}{2})}^{4} - 1}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{{(\frac{1}{2})}^{4} - 1}{\frac{1}{2} - 1}$

Step 9

Simplify the numerator.

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Step 9.1

Apply the product rule to

\frac{1}{2}

$\frac{1}{2}$ .

S_{4} = 20 \cdot \frac{\frac{1^{4}}{2^{4}} - 1}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{1^{4}}{2^{4}} - 1}{\frac{1}{2} - 1}$

Step 9.2

One to any power is one.

S_{4} = 20 \cdot \frac{\frac{1}{2^{4}} - 1}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{1}{2^{4}} - 1}{\frac{1}{2} - 1}$

Step 9.3

Raise

2

$2$ to the power of

4

$4$ .

S_{4} = 20 \cdot \frac{\frac{1}{16} - 1}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{1}{16} - 1}{\frac{1}{2} - 1}$

Step 9.4

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{16}{16}

$\frac{16}{16}$ .

S_{4} = 20 \cdot \frac{\frac{1}{16} - 1 \cdot \frac{16}{16}}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{1}{16} - 1 \cdot \frac{16}{16}}{\frac{1}{2} - 1}$

Step 9.5

Combine

- 1

$- 1$ and

\frac{16}{16}

$\frac{16}{16}$ .

S_{4} = 20 \cdot \frac{\frac{1}{16} + \frac{- 1 \cdot 16}{16}}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{1}{16} + \frac{- 1 \cdot 16}{16}}{\frac{1}{2} - 1}$

Step 9.6

Combine the numerators over the common denominator.

S_{4} = 20 \cdot \frac{\frac{1 - 1 \cdot 16}{16}}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{1 - 1 \cdot 16}{16}}{\frac{1}{2} - 1}$

Step 9.7

Simplify the numerator.

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Step 9.7.1

Multiply

- 1

$- 1$ by

16

$16$ .

S_{4} = 20 \cdot \frac{\frac{1 - 16}{16}}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{1 - 16}{16}}{\frac{1}{2} - 1}$

Step 9.7.2

Subtract

16

$16$ from

1

$1$ .

S_{4} = 20 \cdot \frac{\frac{- 15}{16}}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{- 15}{16}}{\frac{1}{2} - 1}$

S_{4} = 20 \cdot \frac{\frac{- 15}{16}}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{\frac{- 15}{16}}{\frac{1}{2} - 1}$

Step 9.8

Move the negative in front of the fraction.

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} - 1}$

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} - 1}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} - 1}$

Step 10

Simplify the denominator.

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Step 10.1

To write

- 1

$- 1$ as a fraction with a common denominator, multiply by

\frac{2}{2}

$\frac{2}{2}$ .

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} - 1 \cdot \frac{2}{2}}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 10.2

Combine

- 1

$- 1$ and

\frac{2}{2}

$\frac{2}{2}$ .

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 10.3

Combine the numerators over the common denominator.

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1 - 1 \cdot 2}{2}}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1 - 1 \cdot 2}{2}}$

Step 10.4

Simplify the numerator.

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Step 10.4.1

Multiply

- 1

$- 1$ by

2

$2$ .

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1 - 2}{2}}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{1 - 2}{2}}$

Step 10.4.2

Subtract

2

$2$ from

1

$1$ .

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{- 1}{2}}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{- 1}{2}}$

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{- 1}{2}}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{\frac{- 1}{2}}$

Step 10.5

Move the negative in front of the fraction.

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{- \frac{1}{2}}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{- \frac{1}{2}}$

S_{4} = 20 \cdot \frac{- \frac{15}{16}}{- \frac{1}{2}}

$S_{4} = 20 \cdot \frac{- \frac{15}{16}}{- \frac{1}{2}}$

Step 11

Dividing two negative values results in a positive value.

S_{4} = 20 \cdot \frac{\frac{15}{16}}{\frac{1}{2}}

$S_{4} = 20 \cdot \frac{\frac{15}{16}}{\frac{1}{2}}$

Step 12

Multiply the numerator by the reciprocal of the denominator.

S_{4} = 20 \cdot (\frac{15}{16} \cdot 2)

$S_{4} = 20 \cdot (\frac{15}{16} \cdot 2)$

Step 13

Cancel the common factor of

2

$2$ .

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Step 13.1

Factor

2

$2$ out of

16

$16$ .

S_{4} = 20 \cdot (\frac{15}{2 (8)} \cdot 2)

$S_{4} = 20 \cdot (\frac{15}{2 (8)} \cdot 2)$

Step 13.2

Cancel the common factor.

$S_{4} = 20 \cdot (\frac{15}{2 \cdot 8} \cdot 2)$

Step 13.3

Rewrite the expression.

$S_{4} = 20 \cdot \frac{15}{8}$

Step 14

Cancel the common factor of

$4$ .

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Step 14.1

Factor

$4$ out of

$20$ .

$S_{4} = 4 (5) \cdot \frac{15}{8}$

Step 14.2

Factor

$4$ out of

$8$ .

$S_{4} = 4 \cdot 5 \cdot \frac{15}{4 \cdot 2}$

Step 14.3

Cancel the common factor.

$S_{4} = 4 \cdot 5 \cdot \frac{15}{4 \cdot 2}$

Step 14.4

Rewrite the expression.

$S_{4} = 5 \cdot \frac{15}{2}$

Step 15

Combine

$5$ and

$\frac{15}{2}$ .

$S_{4} = \frac{5 \cdot 15}{2}$

Step 16

Multiply

$5$ by

$15$ .

$S_{4} = \frac{75}{2}$