Find the 8th Term 24 , -6 , 3/2 , -3/8 , 3/32 , -3/128

$24$ , $- 6$ , $\frac{3}{2}$ , $- \frac{3}{8}$ , $\frac{3}{32}$ , $- \frac{3}{128}$

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 $- \frac{1}{4}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = - \frac{1}{4}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 24$ and $r = - \frac{1}{4}$ .

$a_{n} = 24 {(- \frac{1}{4})}^{n - 1}$

Step 4

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

Apply the product rule to $- \frac{1}{4}$ .

$a_{n} = 24 ({(- 1)}^{n - 1} {(\frac{1}{4})}^{n - 1})$

Apply the product rule to $\frac{1}{4}$ .

$a_{n} = 24 ({(- 1)}^{n - 1} \frac{1^{n - 1}}{4^{n - 1}})$

Step 5

One to any power is one.

$a_{n} = 24 ({(- 1)}^{n - 1} \frac{1}{4^{n - 1}})$

Step 6

Combine ${(- 1)}^{n - 1}$ and $\frac{1}{4^{n - 1}}$ .

$a_{n} = 24 \frac{{(- 1)}^{n - 1}}{4^{n - 1}}$

Step 7

Combine $24$ and $\frac{{(- 1)}^{n - 1}}{4^{n - 1}}$ .

$a_{n} = \frac{24 {(- 1)}^{n - 1}}{4^{n - 1}}$

Step 8

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

$a_{8} = \frac{24 {(- 1)}^{(8) - 1}}{4^{(8) - 1}}$

Step 9

Simplify the numerator.

Subtract $1$ from $8$ .

$a_{8} = \frac{24 {(- 1)}^{7}}{4^{8 - 1}}$

Raise $- 1$ to the power of $7$ .

$a_{8} = \frac{24 \cdot - 1}{4^{8 - 1}}$

Step 10

Simplify the denominator.

Subtract $1$ from $8$ .

$a_{8} = \frac{24 \cdot - 1}{4^{7}}$

Raise $4$ to the power of $7$ .

$a_{8} = \frac{24 \cdot - 1}{16384}$

Step 11

Reduce the expression by cancelling the common factors.

Multiply $24$ by $- 1$ .

$a_{8} = \frac{- 24}{16384}$

Cancel the common factor of $- 24$ and $16384$ .

Factor $8$ out of $- 24$ .

$a_{8} = \frac{8 (- 3)}{16384}$

Cancel the common factors.

Factor $8$ out of $16384$ .

$a_{8} = \frac{8 \cdot - 3}{8 \cdot 2048}$

Cancel the common factor.

$a_{8} = \frac{8 \cdot - 3}{8 \cdot 2048}$

Rewrite the expression.

$a_{8} = \frac{- 3}{2048}$

Move the negative in front of the fraction.

$a_{8} = - \frac{3}{2048}$

Find the 8th Term 24 , -6 , 3/2 , -3/8 , 3/32 , -3/128

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