Find the 5th Term 2/3 , 1/2 , 3/8 , 9/32

$\frac{2}{3}$ , $\frac{1}{2}$ , $\frac{3}{8}$ , $\frac{9}{32}$

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

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

Step 2

This is the form of a geometric sequence.

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

Step 3

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

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

Step 4

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

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

Step 5

Combine.

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

Step 6

Cancel the common factor of $3^{n - 1}$ and $3$ .

Factor $3$ out of $2 \cdot 3^{n - 1}$ .

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

Cancel the common factors.

Factor $3$ out of $3 \cdot 4^{n - 1}$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 7

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

$a_{5} = \frac{2 \cdot 3^{(5) - 2}}{4^{(5) - 1}}$

Step 8

Simplify the numerator.

Subtract $2$ from $5$ .

$a_{5} = \frac{2 \cdot 3^{3}}{4^{5 - 1}}$

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

$a_{5} = \frac{2 \cdot 27}{4^{5 - 1}}$

Step 9

Simplify the denominator.

Subtract $1$ from $5$ .

$a_{5} = \frac{2 \cdot 27}{4^{4}}$

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

$a_{5} = \frac{2 \cdot 27}{256}$

Step 10

Reduce the expression by cancelling the common factors.

Multiply $2$ by $27$ .

$a_{5} = \frac{54}{256}$

Cancel the common factor of $54$ and $256$ .

Factor $2$ out of $54$ .

$a_{5} = \frac{2 (27)}{256}$

Cancel the common factors.

Factor $2$ out of $256$ .

$a_{5} = \frac{2 \cdot 27}{2 \cdot 128}$

Cancel the common factor.

$a_{5} = \frac{2 \cdot 27}{2 \cdot 128}$

Rewrite the expression.

$a_{5} = \frac{27}{128}$

Find the 5th Term 2/3 , 1/2 , 3/8 , 9/32

Related Questions

Support LCMGCF.com by disabling your adblocker.