Let a (n) be the finite sequence with 9 terms a (1),a(2), ……………….., a(9) defined as follows:
`a(n)={(1{("if the digit n occurs infinitely many times in the decimal expanssion of" 4/3):}),(2{("If the digit n occurs sold number of times in the decmal expansion of" 4/3):}),(3{("If the digit n occurs an even number of times in the decimal expansioni of" 4/3):}):}`
Find all the terms of the sequence.

Answer

To solve the problem, we need to analyze the decimal expansion of \( \frac{4}{3} \) and determine how many times each digit from 1 to 9 appears in that expansion. The sequence \( a(n) \) is defined based on the frequency of each digit \( n \) in the decimal expansion.

### Step-by-Step Solution:

1. **Find the Decimal Expansion of \( \frac{4}{3} \)**:
\[ \frac{4}{3} = 1.33333\ldots \]
The decimal expansion is \( 1.3\overline{3} \), which means it has a repeating digit '3' after the decimal point.

2. **Analyze Each Digit from 1 to 9**:
We will check how many times each digit (1 to 9) appears in the decimal expansion.

- **For \( n = 1 \)**:
- The digit '1' appears **once** (in the whole number part).
- Since it occurs an odd number of times, \( a(1) = 2 \).

- **For \( n = 2 \)**:
- The digit '2' does **not** appear.
- Since it occurs zero times (even), \( a(2) = 3 \).

- **For \( n = 3 \)**:
- The digit '3' appears **infinitely** many times (in the decimal part).
- Therefore, \( a(3) = 1 \).

- **For \( n = 4 \)**:
- The digit '4' does **not** appear.
- Since it occurs zero times (even), \( a(4) = 3 \).

- **For \( n = 5 \)**:
- The digit '5' does **not** appear.
- Since it occurs zero times (even), \( a(5) = 3 \).

- **For \( n = 6 \)**:
- The digit '6' does **not** appear.
- Since it occurs zero times (even), \( a(6) = 3 \).

- **For \( n = 7 \)**:
- The digit '7' does **not** appear.
- Since it occurs zero times (even), \( a(7) = 3 \).

- **For \( n = 8 \)**:
- The digit '8' does **not** appear.
- Since it occurs zero times (even), \( a(8) = 3 \).

- **For \( n = 9 \)**:
- The digit '9' does **not** appear.
- Since it occurs zero times (even), \( a(9) = 3 \).

3. **Compile the Sequence**:
Now we can compile the values of \( a(n) \) for \( n = 1 \) to \( n = 9 \):
\[ a(1) = 2, \quad a(2) = 3, \quad a(3) = 1, \quad a(4) = 3, \quad a(5) = 3, \quad a(6) = 3, \quad a(7) = 3, \quad a(8) = 3, \quad a(9) = 3 \]
Thus, the sequence is:
\[ (2, 3, 1, 3, 3, 3, 3, 3, 3) \]

### Final Answer:
The terms of the sequence \( a(n) \) are:
\[ (2, 3, 1, 3, 3, 3, 3, 3, 3) \]