Precalculus Examples

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

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

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

Step 4

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

Tap for more steps...

Step 4.1

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

Tap for more steps...

Step 4.1.1

Raise $\frac{1}{2}$ to the power of $1$ .

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

Step 4.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.2

Combine the opposite terms in $1 + n - 1$ .

Tap for more steps...

Step 4.2.1

Subtract $1$ from $1$ .

$a_{n} = {(\frac{1}{2})}^{n + 0}$

Step 4.2.2

Add $n$ and $0$ .

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

Step 5

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

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

Step 6

One to any power is one.

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

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 $S_{4}$ .

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

Step 9

Simplify the numerator.

Tap for more steps...

Step 9.1

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

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

Step 9.2

One to any power is one.

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

Step 9.3

Raise $2$ to the power of $4$ .

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

Step 9.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{16}{16}$ .

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

Step 9.5

Combine $- 1$ and $\frac{16}{16}$ .

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

Step 9.6

Combine the numerators over the common denominator.

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

Step 9.7

Simplify the numerator.

Tap for more steps...

Step 9.7.1

Multiply $- 1$ by $16$ .

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

Step 9.7.2

Subtract $16$ from $1$ .

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

Step 9.8

Move the negative in front of the fraction.

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

Step 10

Simplify the denominator.

Tap for more steps...

Step 10.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 10.2

Combine $- 1$ and $\frac{2}{2}$ .

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

Step 10.3

Combine the numerators over the common denominator.

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

Step 10.4

Simplify the numerator.

Tap for more steps...

Step 10.4.1

Multiply $- 1$ by $2$ .

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

Step 10.4.2

Subtract $2$ from $1$ .

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

Step 10.5

Move the negative in front of the fraction.

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

Step 11

Reduce the expression by cancelling the common factors.

Tap for more steps...

Step 11.1

Dividing two negative values results in a positive value.

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

Step 11.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 11.3

Cancel the common factor of $2$ .

Tap for more steps...

Step 11.3.1

Factor $2$ out of $\frac{15}{16} \cdot 2$ .

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

Step 11.3.2

Cancel the common factor.

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

Step 11.3.3

Rewrite the expression.

$S_{4} = \frac{15}{16}$