Algebra Examples

$- 8 + (- 4) + (- 2) + (- 1) + (- \frac{1}{2})$

Step 1

Remove parentheses.

$- 8, - 4, - 2, - 1, - \frac{1}{2}$

Step 2

Remove parentheses.

$- 8, - 4, - 2, - 1, - \frac{1}{2}$

Step 3

Remove parentheses.

$- 8, - 4, - 2, - 1, - \frac{1}{2}$

Step 4

Remove parentheses.

$- 8, - 4, - 2, - 1, - \frac{1}{2}$

Step 5

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 6

This is the form of a geometric sequence.

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

Step 7

Substitute in the values of $a_{1} = - 8$ and $r = \frac{1}{2}$ .

$a_{n} = - 8 {(\frac{1}{2})}^{n - 1}$

Step 8

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

$a_{n} = - 8 \frac{1^{n - 1}}{2^{n - 1}}$

Step 9

One to any power is one.

$a_{n} = - 8 \frac{1}{2^{n - 1}}$

Step 10

Combine $- 8$ and $\frac{1}{2^{n - 1}}$ .

$a_{n} = \frac{- 8}{2^{n - 1}}$

Step 11

Move the negative in front of the fraction.

$a_{n} = - \frac{8}{2^{n - 1}}$

Step 12

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 13

Replace the variables with the known values to find $S_{5}$ .

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

Step 14

Simplify the numerator.

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Step 14.1

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

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

Step 14.2

One to any power is one.

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

Step 14.3

Raise $2$ to the power of $5$ .

$S_{5} = - 8 \cdot \frac{\frac{1}{32} - 1}{\frac{1}{2} - 1}$

Step 14.4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$S_{5} = - 8 \cdot \frac{\frac{1}{32} - 1 \cdot \frac{32}{32}}{\frac{1}{2} - 1}$

Step 14.5

Combine $- 1$ and $\frac{32}{32}$ .

$S_{5} = - 8 \cdot \frac{\frac{1}{32} + \frac{- 1 \cdot 32}{32}}{\frac{1}{2} - 1}$

Step 14.6

Combine the numerators over the common denominator.

$S_{5} = - 8 \cdot \frac{\frac{1 - 1 \cdot 32}{32}}{\frac{1}{2} - 1}$

Step 14.7

Simplify the numerator.

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Step 14.7.1

Multiply $- 1$ by $32$ .

$S_{5} = - 8 \cdot \frac{\frac{1 - 32}{32}}{\frac{1}{2} - 1}$

Step 14.7.2

Subtract $32$ from $1$ .

$S_{5} = - 8 \cdot \frac{\frac{- 31}{32}}{\frac{1}{2} - 1}$

Step 14.8

Move the negative in front of the fraction.

$S_{5} = - 8 \cdot \frac{- \frac{31}{32}}{\frac{1}{2} - 1}$

Step 15

Simplify the denominator.

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Step 15.1

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

$S_{5} = - 8 \cdot \frac{- \frac{31}{32}}{\frac{1}{2} - 1 \cdot \frac{2}{2}}$

Step 15.2

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

$S_{5} = - 8 \cdot \frac{- \frac{31}{32}}{\frac{1}{2} + \frac{- 1 \cdot 2}{2}}$

Step 15.3

Combine the numerators over the common denominator.

$S_{5} = - 8 \cdot \frac{- \frac{31}{32}}{\frac{1 - 1 \cdot 2}{2}}$

Step 15.4

Simplify the numerator.

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Step 15.4.1

Multiply $- 1$ by $2$ .

$S_{5} = - 8 \cdot \frac{- \frac{31}{32}}{\frac{1 - 2}{2}}$

Step 15.4.2

Subtract $2$ from $1$ .

$S_{5} = - 8 \cdot \frac{- \frac{31}{32}}{\frac{- 1}{2}}$

Step 15.5

Move the negative in front of the fraction.

$S_{5} = - 8 \cdot \frac{- \frac{31}{32}}{- \frac{1}{2}}$

Step 16

Dividing two negative values results in a positive value.

$S_{5} = - 8 \cdot \frac{\frac{31}{32}}{\frac{1}{2}}$

Step 17

Multiply the numerator by the reciprocal of the denominator.

$S_{5} = - 8 \cdot (\frac{31}{32} \cdot 2)$

Step 18

Cancel the common factor of $2$ .

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Step 18.1

Factor $2$ out of $32$ .

$S_{5} = - 8 \cdot (\frac{31}{2 (16)} \cdot 2)$

Step 18.2

Cancel the common factor.

$S_{5} = - 8 \cdot (\frac{31}{2 \cdot 16} \cdot 2)$

Step 18.3

Rewrite the expression.

$S_{5} = - 8 \cdot \frac{31}{16}$

Step 19

Cancel the common factor of $8$ .

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Step 19.1

Factor $8$ out of $- 8$ .

$S_{5} = 8 (- 1) \cdot \frac{31}{16}$

Step 19.2

Factor $8$ out of $16$ .

$S_{5} = 8 \cdot - 1 \cdot \frac{31}{8 \cdot 2}$

Step 19.3

Cancel the common factor.

$S_{5} = 8 \cdot - 1 \cdot \frac{31}{8 \cdot 2}$

Step 19.4

Rewrite the expression.

$S_{5} = - 1 \cdot \frac{31}{2}$

Step 20

Rewrite $- 1 (\frac{31}{2})$ as $- (\frac{31}{2})$ .

$S_{5} = - \frac{31}{2}$