Find the Next Term 80 , 20 , 5

$80$ , $20$ , $5$

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 $a_{1} = 80$ and $r = \frac{1}{4}$ .

$a_{n} = 80 {(\frac{1}{4})}^{n - 1}$

Step 4

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

$a_{n} = 80 \frac{1^{n - 1}}{4^{n - 1}}$

Step 5

One to any power is one.

$a_{n} = 80 \frac{1}{4^{n - 1}}$

Step 6

Combine $80$ and $\frac{1}{4^{n - 1}}$ .

$a_{n} = \frac{80}{4^{n - 1}}$

Step 7

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

$a_{4} = \frac{80}{4^{(4) - 1}}$

Step 8

Simplify the denominator.

Subtract $1$ from $4$ .

$a_{4} = \frac{80}{4^{3}}$

Raise $4$ to the power of $3$ .

$a_{4} = \frac{80}{64}$

Step 9

Cancel the common factor of $80$ and $64$ .

Factor $16$ out of $80$ .

$a_{4} = \frac{16 (5)}{64}$

Cancel the common factors.

Factor $16$ out of $64$ .

$a_{4} = \frac{16 \cdot 5}{16 \cdot 4}$

Cancel the common factor.

$a_{4} = \frac{16 \cdot 5}{16 \cdot 4}$

Rewrite the expression.

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

Find the Next Term 80 , 20 , 5

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