Algebra Examples

$81$ , $- 27$ , $9$ , $- 3$ , $1$ , $- \frac{1}{3}$

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 $- \frac{1}{3}$ 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 $a_{1} = 81$ and $r = - \frac{1}{3}$ .

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

Step 4

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

One to any power is one.

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

Step 6

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

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

Step 7

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

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

Step 8

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

$a_{7} = \frac{81 {(- 1)}^{(7) - 1}}{3^{(7) - 1}}$

Step 9

Simplify the numerator.

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Step 9.1

Subtract $1$ from $7$ .

$a_{7} = \frac{81 {(- 1)}^{6}}{3^{7 - 1}}$

Step 9.2

Raise $- 1$ to the power of $6$ .

$a_{7} = \frac{81 \cdot 1}{3^{7 - 1}}$

Step 10

Simplify the denominator.

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Step 10.1

Subtract $1$ from $7$ .

$a_{7} = \frac{81 \cdot 1}{3^{6}}$

Step 10.2

Raise $3$ to the power of $6$ .

$a_{7} = \frac{81 \cdot 1}{729}$

Step 11

Reduce the expression by cancelling the common factors.

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Step 11.1

Multiply $81$ by $1$ .

$a_{7} = \frac{81}{729}$

Step 11.2

Cancel the common factor of $81$ and $729$ .

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Step 11.2.1

Factor $81$ out of $81$ .

$a_{7} = \frac{81 (1)}{729}$

Step 11.2.2

Cancel the common factors.

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Step 11.2.2.1

Factor $81$ out of $729$ .

$a_{7} = \frac{81 \cdot 1}{81 \cdot 9}$

Step 11.2.2.2

Cancel the common factor.

$a_{7} = \frac{81 \cdot 1}{81 \cdot 9}$

Step 11.2.2.3

Rewrite the expression.

$a_{7} = \frac{1}{9}$