Algebra Examples

$\frac{1}{81}$ , $\frac{1}{27}$ , $\frac{1}{9}$ , $\frac{1}{3}$ , $1$ , $3$ , $9$ , $27$ , $81$ , $243$ , $729$

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 $3$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = 3$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{1}{81}$ and $r = 3$ .

$a_{n} = \frac{1}{81} \cdot 3^{n - 1}$

Step 4

Combine $\frac{1}{81}$ and $3^{n - 1}$ .

$a_{n} = \frac{3^{n - 1}}{81}$

Step 5

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

$a_{12} = \frac{3^{(12) - 1}}{81}$

Step 6

Simplify the numerator.

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Step 6.1

Subtract $1$ from $12$ .

$a_{12} = \frac{3^{11}}{81}$

Step 6.2

Raise $3$ to the power of $11$ .

$a_{12} = \frac{177147}{81}$

Step 7

Divide $177147$ by $81$ .

$a_{12} = 2187$