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.