​The first term of a geometric sequence is 2/3. The next three terms are 1/2, 3/8, and 9/32. What is the tenth term of the sequence?

In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio, \( r \). Given the terms:

- First term (\( a \)) = \( \frac{2}{3} \)
- Second term = \( \frac{1}{2} \)
- Third term = \( \frac{3}{8} \)
- Fourth term = \( \frac{9}{32} \)

### Step 1: Finding the Common Ratio, \( r \)
To find the common ratio, divide the second term by the first term:
\[
r = \frac{\frac{1}{2}}{\frac{2}{3}} = \frac{1}{2} \times \frac{3}{2} = \frac{3}{4}
\]

To verify, check the ratio for the third and fourth terms:
\[
r = \frac{\frac{3}{8}}{\frac{1}{2}} = \frac{3}{8} \times 2 = \frac{3}{4}
\]
\[
r = \frac{\frac{9}{32}}{\frac{3}{8}} = \frac{9}{32} \times \frac{8}{3} = \frac{3}{4}
\]

So, the common ratio \( r = \frac{3}{4} \) is consistent.

### Step 2: Finding the Tenth Term
The \( n \)-th term of a geometric sequence is given by:
\[
a_n = a \cdot r^{n-1}
\]
For the tenth term (\( a_{10} \)):
\[
a_{10} = \frac{2}{3} \cdot \left( \frac{3}{4} \right)^{9}
\]

### Step 3: Calculating \( \left( \frac{3}{4} \right)^9 \)
Calculating \( \left( \frac{3}{4} \right)^9 \) gives:
\[
\left( \frac{3}{4} \right)^9 = \frac{19683}{262144}
\]

Thus:
\[
a_{10} = \frac{2}{3} \cdot \frac{19683}{262144} = \frac{39366}{786432}
\]

After simplifying, we get:
\[
a_{10} = \frac{19683}{393216}
\]

So, the tenth term of the sequence is \( \frac{19683}{393216} \).

Would you like further details or have any questions?

### Related Questions

1. How do you find the common ratio in a geometric sequence?
2. How can you determine if a sequence is geometric?
3. What is the formula for the \( n \)-th term of a geometric sequence?
4. How do you calculate higher powers of fractions accurately?
5. How can the sum of a geometric sequence be calculated?

**Tip:** When calculating terms in a geometric sequence, always verify the common ratio across multiple terms to ensure consistency.

The first term of a geometric sequence is 2/3. The next three terms are 1/2, 3/8, and 9/32. What is the tenth term of the sequence?

Learn how to find the tenth term of a geometric sequence where the first term is 2/3 and the next terms are 1/2, 3/8, and 9/32. This solution covers identifying the common ratio, applying the nth term formula, and simplifying fractional powers. Ideal for high school students studying sequences.

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