Find the 5th Term 64 , -16 , 4 , -1

$64$ , $- 16$ , $4$ , $- 1$

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 $a_{1} = 64$ and $r = - \frac{1}{4}$ .

$a_{n} = 64 {(- \frac{1}{4})}^{n - 1}$

Step 4

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

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

$a_{n} = 64 ({(- 1)}^{n - 1} {(\frac{1}{4})}^{n - 1})$

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

$a_{n} = 64 ({(- 1)}^{n - 1} \frac{1^{n - 1}}{4^{n - 1}})$

Step 5

One to any power is one.

$a_{n} = 64 ({(- 1)}^{n - 1} \frac{1}{4^{n - 1}})$

Step 6

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

$a_{n} = 64 \frac{{(- 1)}^{n - 1}}{4^{n - 1}}$

Step 7

Combine $64$ and $\frac{{(- 1)}^{n - 1}}{4^{n - 1}}$ .

$a_{n} = \frac{64 {(- 1)}^{n - 1}}{4^{n - 1}}$

Step 8

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

$a_{5} = \frac{64 {(- 1)}^{(5) - 1}}{4^{(5) - 1}}$

Step 9

Simplify the numerator.

Subtract $1$ from $5$ .

$a_{5} = \frac{64 {(- 1)}^{4}}{4^{5 - 1}}$

Raise $- 1$ to the power of $4$ .

$a_{5} = \frac{64 \cdot 1}{4^{5 - 1}}$

Step 10

Simplify the denominator.

Subtract $1$ from $5$ .

$a_{5} = \frac{64 \cdot 1}{4^{4}}$

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

$a_{5} = \frac{64 \cdot 1}{256}$

Step 11

Reduce the expression by cancelling the common factors.

Multiply $64$ by $1$ .

$a_{5} = \frac{64}{256}$

Cancel the common factor of $64$ and $256$ .

Factor $64$ out of $64$ .

$a_{5} = \frac{64 (1)}{256}$

Cancel the common factors.

Factor $64$ out of $256$ .

$a_{5} = \frac{64 \cdot 1}{64 \cdot 4}$

Cancel the common factor.

$a_{5} = \frac{64 \cdot 1}{64 \cdot 4}$

Rewrite the expression.

$a_{5} = \frac{1}{4}$

Find the 5th Term 64 , -16 , 4 , -1

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