Algebra 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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

One to any power is one.

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

Step 6

Combine ${(- 1)}^{n - 1}$ and $\frac{1}{2^{n - 1}}$ .

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

Step 7

Combine.

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

Step 8

Multiply $2$ by $2^{n - 1}$ by adding the exponents.

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Step 8.1

Multiply $2$ by $2^{n - 1}$ .

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Step 8.1.1

Raise $2$ to the power of $1$ .

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

Step 8.1.2

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

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

Step 8.2

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

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Step 8.2.1

Subtract $1$ from $1$ .

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

Step 8.2.2

Add $n$ and $0$ .

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

Step 9

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

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

Step 10

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 11

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 12

Simplify the numerator.

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Step 12.1

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 12.1.1

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

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

Step 12.1.2

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

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

Step 12.2

Raise $- 1$ to the power of $4$ .

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

Step 12.3

Multiply $\frac{1^{4}}{2^{4}}$ by $1$ .

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

Step 12.4

One to any power is one.

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

Step 12.5

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

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

Step 12.6

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 12.7

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 12.8

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 12.9

Simplify the numerator.

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Step 12.9.1

Multiply $- 1$ by $16$ .

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

Step 12.9.2

Subtract $16$ from $1$ .

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

Step 12.10

Move the negative in front of the fraction.

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

Step 13

Simplify the denominator.

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Step 13.1

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 13.2

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 13.3

Combine the numerators over the common denominator.

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

Step 13.4

Simplify the numerator.

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Step 13.4.1

Multiply $- 1$ by $2$ .

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

Step 13.4.2

Subtract $2$ from $- 1$ .

$S_{4} = \frac{1}{2} \cdot \frac{- \frac{15}{16}}{\frac{- 3}{2}}$

Step 13.5

Move the negative in front of the fraction.

$S_{4} = \frac{1}{2} \cdot \frac{- \frac{15}{16}}{- \frac{3}{2}}$

Step 14

Dividing two negative values results in a positive value.

$S_{4} = \frac{1}{2} \cdot \frac{\frac{15}{16}}{\frac{3}{2}}$

Step 15

Multiply the numerator by the reciprocal of the denominator.

$S_{4} = \frac{1}{2} \cdot (\frac{15}{16} \cdot \frac{2}{3})$

Step 16

Cancel the common factor of $3$ .

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Step 16.1

Factor $3$ out of $15$ .

$S_{4} = \frac{1}{2} \cdot (\frac{3 (5)}{16} \cdot \frac{2}{3})$

Step 16.2

Cancel the common factor.

$S_{4} = \frac{1}{2} \cdot (\frac{3 \cdot 5}{16} \cdot \frac{2}{3})$

Step 16.3

Rewrite the expression.

$S_{4} = \frac{1}{2} \cdot (\frac{5}{16} \cdot 2)$

Step 17

Cancel the common factor of $2$ .

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Step 17.1

Factor $2$ out of $16$ .

$S_{4} = \frac{1}{2} \cdot (\frac{5}{2 (8)} \cdot 2)$

Step 17.2

Cancel the common factor.

$S_{4} = \frac{1}{2} \cdot (\frac{5}{2 \cdot 8} \cdot 2)$

Step 17.3

Rewrite the expression.

$S_{4} = \frac{1}{2} \cdot \frac{5}{8}$

Step 18

Multiply $\frac{1}{2} \cdot \frac{5}{8}$ .

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Step 18.1

Multiply $\frac{1}{2}$ by $\frac{5}{8}$ .

$S_{4} = \frac{5}{2 \cdot 8}$

Step 18.2

Multiply $2$ by $8$ .

$S_{4} = \frac{5}{16}$