We are tasked with solving for $a_n$ and $a_5$ in the given geometric sequence $\frac{9}{5}, 3, 5, \frac{25}{3}$. Let's proceed step by step.

Step 1: Identify the common ratio $r$

To find $r$, divide any term by the previous term in the geometric sequence:

1. Between the 2nd and 1st terms: $r = \frac{3}{\frac{9}{5}} = 3 \times \frac{5}{9} = \frac{15}{9} = \frac{5}{3}.$

2. Between the 3rd and 2nd terms: r = \frac{5}{3} = \frac{5}{3} \quad (\text{same ratio confirms r}).

Thus, the common ratio $r$ is $\frac{5}{3}$.

Step 2: Write the general term $a_n$

The formula for the $n$-th term of a geometric sequence is: $a_n = a_1 \cdot r^{n-1},$ where $a_1 = \frac{9}{5}$ and $r = \frac{5}{3}$. Substituting these values, we get: $a_n = \frac{9}{5} \cdot \left( \frac{5}{3} \right)^{n-1}.$

Step 3: Simplify the expression for $a_n$

Simplify the general term $a_n$ using properties of exponents: $a_n = \frac{9}{5} \cdot \left( \frac{5}{3} \right)^{n-1}.$

• Notice that $\left( \frac{5}{3} \right)^{n-1}$ stays as a power.

Thus, the formula for $a_n$ is: $a_n = \frac{9}{5} \cdot \left( \frac{5}{3} \right)^{n-1}.$

Step 4: Find $a_5$

To find $a_5$, substitute $n = 5$ into $a_n$:

$a_5 = \frac{9}{5} \cdot \left( \frac{5}{3} \right)^{5-1} = \frac{9}{5} \cdot \left( \frac{5}{3} \right)^4.$

Simplify $\left( \frac{5}{3} \right)^4$: $\left( \frac{5}{3} \right)^4 = \frac{5^4}{3^4} = \frac{625}{81}.$

Now multiply: $a_5 = \frac{9}{5} \cdot \frac{625}{81}.$ Simplify step-by-step: $a_5 = \frac{9 \cdot 625}{5 \cdot 81} = \frac{5625}{405}.$

Simplify the fraction: $a_5 = \frac{125}{9}.$

Final Answers:

1. General term $a_n$: $a_n = \frac{9}{5} \cdot \left( \frac{5}{3} \right)^{n-1}.$
2. Fifth term $a_5$: $a_5 = \frac{125}{9}.$

Do you need further clarification or additional details? Here are 5 related questions:

1. How do you determine the common ratio in a geometric sequence?
2. What is the formula for the sum of a geometric sequence?
3. How would the formula change if the common ratio were negative?
4. Can you identify whether a sequence is geometric or arithmetic?
5. How do you find the $n$-th term if the starting term changes?

Tip: Always check for consistency by verifying the ratio between consecutive terms when dealing with geometric sequences.