Algebra Examples

$\frac{27}{16}$ , $- \frac{9}{4}$ , $3$ ,

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 $- \frac{4}{3}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = - \frac{4}{3}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = \frac{27}{16}$ and $r = - \frac{4}{3}$ .

$a_{n} = \frac{27}{16} {(- \frac{4}{3})}^{n - 1}$

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 $- \frac{4}{3}$ .

$a_{n} = \frac{27}{16} ({(- 1)}^{n - 1} {(\frac{4}{3})}^{n - 1})$

Step 4.2

Apply the product rule to $\frac{4}{3}$ .

$a_{n} = \frac{27}{16} ({(- 1)}^{n - 1} \frac{4^{n - 1}}{3^{n - 1}})$

Step 5

Multiply ${(- 1)}^{n - 1} \frac{4^{n - 1}}{3^{n - 1}}$ .

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Step 5.1

Combine ${(- 1)}^{n - 1}$ and $\frac{4^{n - 1}}{3^{n - 1}}$ .

$a_{n} = \frac{27}{16} \cdot \frac{{(- 1)}^{n - 1} \cdot 4^{n - 1}}{3^{n - 1}}$

Step 5.2

Rewrite ${(- 1)}^{n - 1} \cdot 4^{n - 1}$ as ${(- 1 \cdot 4)}^{n - 1}$ .

$a_{n} = \frac{27}{16} \cdot \frac{{(- 1 \cdot 4)}^{n - 1}}{3^{n - 1}}$

Step 5.3

Multiply $- 1$ by $4$ .

$a_{n} = \frac{27}{16} \cdot \frac{{(- 4)}^{n - 1}}{3^{n - 1}}$

Step 6

Combine.

$a_{n} = \frac{27 {(- 4)}^{n - 1}}{16 \cdot 3^{n - 1}}$

Step 7

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

$a_{4} = \frac{27 {(- 4)}^{(4) - 1}}{16 \cdot 3^{(4) - 1}}$

Step 8

Simplify the numerator.

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Step 8.1

Subtract $1$ from $4$ .

$a_{4} = \frac{27 {(- 4)}^{3}}{16 \cdot 3^{4 - 1}}$

Step 8.2

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

$a_{4} = \frac{27 \cdot - 64}{16 \cdot 3^{4 - 1}}$

Step 9

Simplify the denominator.

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Step 9.1

Subtract $1$ from $4$ .

$a_{4} = \frac{27 \cdot - 64}{16 \cdot 3^{3}}$

Step 9.2

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

$a_{4} = \frac{27 \cdot - 64}{16 \cdot 27}$

Step 10

Simplify the expression.

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Step 10.1

Multiply $27$ by $- 64$ .

$a_{4} = \frac{- 1728}{16 \cdot 27}$

Step 10.2

Multiply $16$ by $27$ .

$a_{4} = \frac{- 1728}{432}$

Step 10.3

Divide $- 1728$ by $432$ .

$a_{4} = - 4$