Find the 6th Term -2 , -1 , -1/2 , -1/4

$- 2$ , $- 1$ , $- \frac{1}{2}$ , $- \frac{1}{4}$

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

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

Step 4

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

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

Step 5

One to any power is one.

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

Step 6

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

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

Step 7

Move the negative in front of the fraction.

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

Step 8

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

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

Step 9

Simplify the denominator.

Subtract $1$ from $6$ .

$a_{6} = - \frac{2}{2^{5}}$

Raise $2$ to the power of $5$ .

$a_{6} = - \frac{2}{32}$

Step 10

Cancel the common factor of $2$ and $32$ .

Factor $2$ out of $2$ .

$a_{6} = - \frac{2 (1)}{32}$

Cancel the common factors.

Factor $2$ out of $32$ .

$a_{6} = - \frac{2 \cdot 1}{2 \cdot 16}$

Cancel the common factor.

$a_{6} = - \frac{2 \cdot 1}{2 \cdot 16}$

Rewrite the expression.

$a_{6} = - \frac{1}{16}$

Find the 6th Term -2 , -1 , -1/2 , -1/4

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