Algebra Examples

$4 + (- 12) + 36 + (- 108)$

Step 1

Remove parentheses.

$4, - 12, 36, - 108$

Step 2

Remove parentheses.

$4, - 12, 36, - 108$

Step 3

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

Geometric Sequence: $r = - 3$

Step 4

This is the form of a geometric sequence.

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

Step 5

Substitute in the values of $a_{1} = 4$ and $r = - 3$ .

$a_{n} = 4 {(- 3)}^{n - 1}$

Step 6

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 7

Replace the variables with the known values to find $S_{4}$ .

$S_{4} = 4 \cdot \frac{{(- 3)}^{4} - 1}{- 3 - 1}$

Step 8

Simplify the numerator.

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Raise $- 3$ to the power of $4$ .

$S_{4} = 4 \cdot \frac{81 - 1}{- 3 - 1}$

Subtract $1$ from $81$ .

$S_{4} = 4 \cdot \frac{80}{- 3 - 1}$

Step 9

Reduce the expression by cancelling the common factors.

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Subtract $1$ from $- 3$ .

$S_{4} = 4 \cdot \frac{80}{- 4}$

Cancel the common factor of $4$ .

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Factor $4$ out of $- 4$ .

$S_{4} = 4 \cdot \frac{80}{4 (- 1)}$

Cancel the common factor.

$S_{4} = 4 \cdot \frac{80}{4 \cdot - 1}$

Rewrite the expression.

$S_{4} = \frac{80}{- 1}$

Divide $80$ by $- 1$ .

$S_{4} = - 80$