Find the Fourth Term 6 , -3 , 3/2

$6$ , $- 3$ , $\frac{3}{2}$

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 $- \frac{1}{2}$ 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 $a_{1} = 6$ and $r = - \frac{1}{2}$ .

$a_{n} = 6 {(- \frac{1}{2})}^{n - 1}$

Step 4

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

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

$a_{n} = 6 ({(- 1)}^{n - 1} {(\frac{1}{2})}^{n - 1})$

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

$a_{n} = 6 ({(- 1)}^{n - 1} \frac{1^{n - 1}}{2^{n - 1}})$

Step 5

One to any power is one.

$a_{n} = 6 ({(- 1)}^{n - 1} \frac{1}{2^{n - 1}})$

Step 6

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

$a_{n} = 6 \frac{{(- 1)}^{n - 1}}{2^{n - 1}}$

Step 7

Combine $6$ and $\frac{{(- 1)}^{n - 1}}{2^{n - 1}}$ .

$a_{n} = \frac{6 {(- 1)}^{n - 1}}{2^{n - 1}}$

Step 8

Substitute in the value of $n$ to find the $n$ th term.

$a_{4} = \frac{6 {(- 1)}^{(4) - 1}}{2^{(4) - 1}}$

Step 9

Simplify the numerator.

Subtract $1$ from $4$ .

$a_{4} = \frac{6 {(- 1)}^{3}}{2^{4 - 1}}$

Raise $- 1$ to the power of $3$ .

$a_{4} = \frac{6 \cdot - 1}{2^{4 - 1}}$

Step 10

Simplify the denominator.

Subtract $1$ from $4$ .

$a_{4} = \frac{6 \cdot - 1}{2^{3}}$

Raise $2$ to the power of $3$ .

$a_{4} = \frac{6 \cdot - 1}{8}$

Step 11

Reduce the expression by cancelling the common factors.

Multiply $6$ by $- 1$ .

$a_{4} = \frac{- 6}{8}$

Cancel the common factor of $- 6$ and $8$ .

Factor $2$ out of $- 6$ .

$a_{4} = \frac{2 (- 3)}{8}$

Cancel the common factors.

Factor $2$ out of $8$ .

$a_{4} = \frac{2 \cdot - 3}{2 \cdot 4}$

Cancel the common factor.

$a_{4} = \frac{2 \cdot - 3}{2 \cdot 4}$

Rewrite the expression.

$a_{4} = \frac{- 3}{4}$

Move the negative in front of the fraction.

$a_{4} = - \frac{3}{4}$

Find the Fourth Term 6 , -3 , 3/2

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