Algebra Examples

$\frac{3}{4}$ , $\frac{1}{2}$ , $\frac{1}{3}$ ,

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

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

This is the form of a geometric sequence.

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

Substitute in the values of $a_{1} = \frac{3}{4}$ and $r = \frac{2}{3}$ .

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

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

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

Combine.

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

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

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

Simplify the numerator.

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Subtract $1$ from $4$ .

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

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

$a_{4} = \frac{3 \cdot 8}{4 \cdot 3^{4 - 1}}$

Simplify the denominator.

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Subtract $1$ from $4$ .

$a_{4} = \frac{3 \cdot 8}{4 \cdot 3^{3}}$

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

$a_{4} = \frac{3 \cdot 8}{4 \cdot 27}$

Reduce the expression by cancelling the common factors.

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Multiply $3$ by $8$ .

$a_{4} = \frac{24}{4 \cdot 27}$

Multiply $4$ by $27$ .

$a_{4} = \frac{24}{108}$

Cancel the common factor of $24$ and $108$ .

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Factor $12$ out of $24$ .

$a_{4} = \frac{12 (2)}{108}$

Cancel the common factors.

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Factor $12$ out of $108$ .

$a_{4} = \frac{12 \cdot 2}{12 \cdot 9}$

Cancel the common factor.

$a_{4} = \frac{12 \cdot 2}{12 \cdot 9}$

Rewrite the expression.

$a_{4} = \frac{2}{9}$