Algebra Examples

$405$ , $135$ , $45$

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

Geometric Sequence: $r = \frac{1}{3}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 405$ and $r = \frac{1}{3}$ .

$a_{n} = 405 {(\frac{1}{3})}^{n - 1}$

Step 4

Apply the product rule to $\frac{1}{3}$ .

$a_{n} = 405 \frac{1^{n - 1}}{3^{n - 1}}$

Step 5

One to any power is one.

$a_{n} = 405 \frac{1}{3^{n - 1}}$

Step 6

Combine $405$ and $\frac{1}{3^{n - 1}}$ .

$a_{n} = \frac{405}{3^{n - 1}}$

Step 7

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

$a_{4} = \frac{405}{3^{(4) - 1}}$

Step 8

Simplify the denominator.

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Step 8.1

Subtract $1$ from $4$ .

$a_{4} = \frac{405}{3^{3}}$

Step 8.2

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

$a_{4} = \frac{405}{27}$

Step 9

Divide $405$ by $27$ .

$a_{4} = 15$