To find the tenth term of the given geometric sequence, we can use the general formula for the $n$-th term of a geometric sequence: $a_n = a \cdot r^{n-1}$ where:

• $a$ is the first term of the sequence,
• $r$ is the common ratio, and
• $n$ is the term number.

Step-by-Step Solution

1. Identify the First Term ($a$): The first term $a$ is given as $\frac{2}{3}$.

2. Determine the Common Ratio ($r$): To find the common ratio, divide the second term by the first term: $r = \frac{\frac{1}{2}}{\frac{2}{3}} = \frac{1}{2} \cdot \frac{3}{2} = \frac{3}{4}$ So, the common ratio $r$ is $\frac{3}{4}$.

3. Find the Tenth Term ($a_{10}$): Substitute $a = \frac{2}{3}$, $r = \frac{3}{4}$, and $n = 10$ into the formula: $a_{10} = \frac{2}{3} \cdot \left( \frac{3}{4} \right)^{10-1}$ $= \frac{2}{3} \cdot \left( \frac{3}{4} \right)^9$

4. Calculate $\left( \frac{3}{4} \right)^9$ and Multiply: To find the exact value, you can compute $\left( \frac{3}{4} \right)^9$ and then multiply by $\frac{2}{3}$, or approximate it using a calculator.

However, expressing $a_{10}$ as: $a_{10} = \frac{2}{3} \cdot \left( \frac{3}{4} \right)^9$ is often sufficient in exact form unless further simplification or a decimal approximation is needed.

Would you like the decimal approximation for the value or more details on any steps?

Here are some related questions:

1. How would the sequence behave if the common ratio was greater than 1?
2. What would the 15th term be in this sequence?
3. How do you determine the sum of the first $n$ terms of a geometric sequence?
4. How does the common ratio affect whether the sequence converges or diverges?
5. What is the limit of the terms of the sequence as $n \to \infty$ in this case?

Tip: In geometric sequences, a common ratio between 0 and 1 leads to terms that decrease in magnitude, while a ratio greater than 1 increases the magnitude of terms.