Algebra Examples

$\frac{- 9}{1024}$ , $\frac{9}{256}$ , $\frac{- 9}{64}$

Step 1

Move the negative in front of the fraction.

$- \frac{9}{1024}, \frac{9}{256}, \frac{- 9}{64}$

Step 2

Move the negative in front of the fraction.

$- \frac{9}{1024}, \frac{9}{256}, - \frac{9}{64}$

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

Geometric Sequence: $r = - 4$

Step 4

This is the form of a geometric sequence.

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

Step 5

Substitute in the values of $a_{1} = - \frac{9}{1024}$ and $r = - 4$ .

$a_{n} = - \frac{9}{1024} {(- 4)}^{n - 1}$

Step 6

Combine ${(- 4)}^{n - 1}$ and $\frac{9}{1024}$ .

$a_{n} = - \frac{{(- 4)}^{n - 1} \cdot 9}{1024}$

Step 7

Move $9$ to the left of ${(- 4)}^{n - 1}$ .

$a_{n} = - \frac{9 {(- 4)}^{n - 1}}{1024}$

Step 8

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

$a_{5} = - \frac{9 {(- 4)}^{(5) - 1}}{1024}$

Step 9

Simplify the numerator.

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Step 9.1

Subtract $1$ from $5$ .

$a_{5} = - \frac{9 {(- 4)}^{4}}{1024}$

Step 9.2

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

$a_{5} = - \frac{9 \cdot 256}{1024}$

Step 10

Reduce the expression by cancelling the common factors.

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Step 10.1

Multiply $9$ by $256$ .

$a_{5} = - \frac{2304}{1024}$

Step 10.2

Cancel the common factor of $2304$ and $1024$ .

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Step 10.2.1

Factor $256$ out of $2304$ .

$a_{5} = - \frac{256 (9)}{1024}$

Step 10.2.2

Cancel the common factors.

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Step 10.2.2.1

Factor $256$ out of $1024$ .

$a_{5} = - \frac{256 \cdot 9}{256 \cdot 4}$

Step 10.2.2.2

Cancel the common factor.

$a_{5} = - \frac{256 \cdot 9}{256 \cdot 4}$

Step 10.2.2.3

Rewrite the expression.

$a_{5} = - \frac{9}{4}$