Find the 7th Term 100 , 20 , 4 , 4/5 , 4/25

$100$ , $20$ , $4$ , $\frac{4}{5}$ , $\frac{4}{25}$

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 $a_{1} = 100$ and $r = \frac{1}{5}$ .

$a_{n} = 100 {(\frac{1}{5})}^{n - 1}$

Step 4

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

$a_{n} = 100 \frac{1^{n - 1}}{5^{n - 1}}$

Step 5

One to any power is one.

$a_{n} = 100 \frac{1}{5^{n - 1}}$

Step 6

Combine $100$ and $\frac{1}{5^{n - 1}}$ .

$a_{n} = \frac{100}{5^{n - 1}}$

Step 7

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

$a_{7} = \frac{100}{5^{(7) - 1}}$

Step 8

Simplify the denominator.

Subtract $1$ from $7$ .

$a_{7} = \frac{100}{5^{6}}$

Raise $5$ to the power of $6$ .

$a_{7} = \frac{100}{15625}$

Step 9

Cancel the common factor of $100$ and $15625$ .

Factor $25$ out of $100$ .

$a_{7} = \frac{25 (4)}{15625}$

Cancel the common factors.

Factor $25$ out of $15625$ .

$a_{7} = \frac{25 \cdot 4}{25 \cdot 625}$

Cancel the common factor.

$a_{7} = \frac{25 \cdot 4}{25 \cdot 625}$

Rewrite the expression.

$a_{7} = \frac{4}{625}$

Find the 7th Term 100 , 20 , 4 , 4/5 , 4/25

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