Find the 6th Term -375 , -75 , -15 , -3

$- 375$ , $- 75$ , $- 15$ , $- 3$

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 $\frac{1}{5}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{1}{5}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = - 375$ and $r = \frac{1}{5}$ .

$a_{n} = - 375 {(\frac{1}{5})}^{n - 1}$

Step 4

Apply the product rule to $\frac{1}{5}$ .

$a_{n} = - 375 \frac{1^{n - 1}}{5^{n - 1}}$

Step 5

One to any power is one.

$a_{n} = - 375 \frac{1}{5^{n - 1}}$

Step 6

Combine $- 375$ and $\frac{1}{5^{n - 1}}$ .

$a_{n} = \frac{- 375}{5^{n - 1}}$

Step 7

Move the negative in front of the fraction.

$a_{n} = - \frac{375}{5^{n - 1}}$

Step 8

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

$a_{6} = - \frac{375}{5^{(6) - 1}}$

Step 9

Simplify the denominator.

Subtract $1$ from $6$ .

$a_{6} = - \frac{375}{5^{5}}$

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

$a_{6} = - \frac{375}{3125}$

Step 10

Cancel the common factor of $375$ and $3125$ .

Factor $125$ out of $375$ .

$a_{6} = - \frac{125 (3)}{3125}$

Cancel the common factors.

Factor $125$ out of $3125$ .

$a_{6} = - \frac{125 \cdot 3}{125 \cdot 25}$

Cancel the common factor.

$a_{6} = - \frac{125 \cdot 3}{125 \cdot 25}$

Rewrite the expression.

$a_{6} = - \frac{3}{25}$

Find the 6th Term -375 , -75 , -15 , -3

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