QuestionThe first term of a geometric sequence is $\frac{2}{3}$ . The next three terms are $\frac{1}{2}, \frac{3}{8}$ , and $\frac{9}{32}$ . What is the tenth term of the sequence?

Studdy Solution

STEP 1

1. The sequence is geometric, meaning each term is obtained by multiplying the previous term by a constant ratio.
2. The first term of the sequence is $\frac{2}{3}$ .
3. The next three terms are $\frac{1}{2}, \frac{3}{8}, \frac{9}{32}$ .
4. We need to find the tenth term of the sequence.

STEP 2

1. Identify the common ratio of the geometric sequence.
2. Use the formula for the $n$ -th term of a geometric sequence.
3. Calculate the tenth term using the identified common ratio and the formula.

STEP 3

Identify the common ratio of the geometric sequence.
To find the common ratio $r$ , divide the second term by the first term:
$r = \frac{\frac{1}{2}}{\frac{2}{3}} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}$
Verify with the third term:
$r = \frac{\frac{3}{8}}{\frac{1}{2}} = \frac{3}{8} \times \frac{2}{1} = \frac{3}{4}$
And the fourth term:
$r = \frac{\frac{9}{32}}{\frac{3}{8}} = \frac{9}{32} \times \frac{8}{3} = \frac{3}{4}$
The common ratio is $\frac{3}{4}$ .

STEP 4

Use the formula for the $n$ -th term of a geometric sequence.
The formula for the $n$ -th term $a_n$ of a geometric sequence is:
$a_n = a_1 \cdot r^{n-1}$
where $a_1$ is the first term and $r$ is the common ratio.

STEP 5

Calculate the tenth term using the identified common ratio and the formula.
Substitute the known values into the formula:
$a_{10} = \frac{2}{3} \cdot \left(\frac{3}{4}\right)^{10-1}$
$a_{10} = \frac{2}{3} \cdot \left(\frac{3}{4}\right)^9$
Calculate $\left(\frac{3}{4}\right)^9$ :
$\left(\frac{3}{4}\right)^9 = \frac{3^9}{4^9} = \frac{19683}{262144}$
Now calculate $a_{10}$ :
$a_{10} = \frac{2}{3} \cdot \frac{19683}{262144} = \frac{2 \times 19683}{3 \times 262144}$
$a_{10} = \frac{39366}{786432}$
Simplify the fraction:
$a_{10} = \frac{39366}{786432} = \frac{1}{20}$
The tenth term of the sequence is:
$\boxed{\frac{1}{20}}$

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms