Algebra Examples

$\frac{7}{9}$ , $- \frac{7}{3}$ , $7$ , $- 21$ , $63$

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 $a_{1} = \frac{7}{9}$ and $r = - 3$ .

$a_{n} = \frac{7}{9} {(- 3)}^{n - 1}$

Step 4

Combine $\frac{7}{9}$ and ${(- 3)}^{n - 1}$ .

$a_{n} = \frac{7 {(- 3)}^{n - 1}}{9}$

Step 5

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

$a_{7} = \frac{7 {(- 3)}^{(7) - 1}}{9}$

Step 6

Simplify the numerator.

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Step 6.1

Subtract $1$ from $7$ .

$a_{7} = \frac{7 {(- 3)}^{6}}{9}$

Step 6.2

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

$a_{7} = \frac{7 \cdot 729}{9}$

Step 7

Simplify the expression.

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Step 7.1

Multiply $7$ by $729$ .

$a_{7} = \frac{5103}{9}$

Step 7.2

Divide $5103$ by $9$ .

$a_{7} = 567$