Find the 7th Term -48 , 12 , -3 , 3/4 , -3/16

$- 48$ , $12$ , $- 3$ , $\frac{3}{4}$ , $- \frac{3}{16}$

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 $a_{1} = - 48$ and $r = - \frac{1}{4}$ .

$a_{n} = - 48 {(- \frac{1}{4})}^{n - 1}$

Step 4

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

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

$a_{n} = - 48 ({(- 1)}^{n - 1} {(\frac{1}{4})}^{n - 1})$

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

$a_{n} = - 48 ({(- 1)}^{n - 1} \frac{1^{n - 1}}{4^{n - 1}})$

Step 5

One to any power is one.

$a_{n} = - 48 ({(- 1)}^{n - 1} \frac{1}{4^{n - 1}})$

Step 6

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

$a_{n} = - 48 \frac{{(- 1)}^{n - 1}}{4^{n - 1}}$

Step 7

Combine $- 48$ and $\frac{{(- 1)}^{n - 1}}{4^{n - 1}}$ .

$a_{n} = \frac{- 48 {(- 1)}^{n - 1}}{4^{n - 1}}$

Step 8

Move the negative in front of the fraction.

$a_{n} = - \frac{48 {(- 1)}^{n - 1}}{4^{n - 1}}$

Step 9

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

$a_{7} = - \frac{48 {(- 1)}^{(7) - 1}}{4^{(7) - 1}}$

Step 10

Simplify the numerator.

Subtract $1$ from $7$ .

$a_{7} = - \frac{48 {(- 1)}^{6}}{4^{7 - 1}}$

Raise $- 1$ to the power of $6$ .

$a_{7} = - \frac{48 \cdot 1}{4^{7 - 1}}$

Step 11

Simplify the denominator.

Subtract $1$ from $7$ .

$a_{7} = - \frac{48 \cdot 1}{4^{6}}$

Raise $4$ to the power of $6$ .

$a_{7} = - \frac{48 \cdot 1}{4096}$

Step 12

Reduce the expression by cancelling the common factors.

Multiply $48$ by $1$ .

$a_{7} = - \frac{48}{4096}$

Cancel the common factor of $48$ and $4096$ .

Factor $16$ out of $48$ .

$a_{7} = - \frac{16 (3)}{4096}$

Cancel the common factors.

Factor $16$ out of $4096$ .

$a_{7} = - \frac{16 \cdot 3}{16 \cdot 256}$

Cancel the common factor.

$a_{7} = - \frac{16 \cdot 3}{16 \cdot 256}$

Rewrite the expression.

$a_{7} = - \frac{3}{256}$

Find the 7th Term -48 , 12 , -3 , 3/4 , -3/16

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