Algebra Examples

$20$ , $30$ , $45$

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 $\frac{3}{2}$ gives the next term. In other words, $a_{n} = a_{1} r^{n - 1}$ .

Geometric Sequence: $r = \frac{3}{2}$

Step 2

This is the form of a geometric sequence.

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

Step 3

Substitute in the values of $a_{1} = 20$ and $r = \frac{3}{2}$ .

$a_{n} = 20 {(\frac{3}{2})}^{n - 1}$

Step 4

Apply the product rule to $\frac{3}{2}$ .

$a_{n} = 20 \frac{3^{n - 1}}{2^{n - 1}}$

Step 5

Combine $20$ and $\frac{3^{n - 1}}{2^{n - 1}}$ .

$a_{n} = \frac{20 \cdot 3^{n - 1}}{2^{n - 1}}$

Step 6

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

$a_{4} = \frac{20 \cdot 3^{(4) - 1}}{2^{(4) - 1}}$

Step 7

Simplify the numerator.

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Step 7.1

Subtract $1$ from $4$ .

$a_{4} = \frac{20 \cdot 3^{3}}{2^{4 - 1}}$

Step 7.2

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

$a_{4} = \frac{20 \cdot 27}{2^{4 - 1}}$

Step 8

Simplify the denominator.

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Step 8.1

Subtract $1$ from $4$ .

$a_{4} = \frac{20 \cdot 27}{2^{3}}$

Step 8.2

Raise $2$ to the power of $3$ .

$a_{4} = \frac{20 \cdot 27}{8}$

Step 9

Reduce the expression by cancelling the common factors.

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Step 9.1

Multiply $20$ by $27$ .

$a_{4} = \frac{540}{8}$

Step 9.2

Cancel the common factor of $540$ and $8$ .

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Step 9.2.1

Factor $4$ out of $540$ .

$a_{4} = \frac{4 (135)}{8}$

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $4$ out of $8$ .

$a_{4} = \frac{4 \cdot 135}{4 \cdot 2}$

Step 9.2.2.2

Cancel the common factor.

$a_{4} = \frac{4 \cdot 135}{4 \cdot 2}$

Step 9.2.2.3

Rewrite the expression.

$a_{4} = \frac{135}{2}$