Let the sequence be defined as follow:
`a_(1)=3`
`a_(n)=3a_(n-1)+2,` for all `n gt 1`.
Find the first five terms of the sequence.

Answer

To find the first five terms of the sequence defined by \( a_1 = 3 \) and \( a_n = 3a_{n-1} + 2 \) for all \( n > 1 \), we will calculate each term step by step.

### Step 1: Calculate \( a_1 \)
Given:
\[ a_1 = 3 \]

### Step 2: Calculate \( a_2 \)
Using the formula for \( n = 2 \):
\[ a_2 = 3a_{2-1} + 2 = 3a_1 + 2 \]
Substituting \( a_1 = 3 \):
\[ a_2 = 3 \times 3 + 2 = 9 + 2 = 11 \]

### Step 3: Calculate \( a_3 \)
Using the formula for \( n = 3 \):
\[ a_3 = 3a_{3-1} + 2 = 3a_2 + 2 \]
Substituting \( a_2 = 11 \):
\[ a_3 = 3 \times 11 + 2 = 33 + 2 = 35 \]

### Step 4: Calculate \( a_4 \)
Using the formula for \( n = 4 \):
\[ a_4 = 3a_{4-1} + 2 = 3a_3 + 2 \]
Substituting \( a_3 = 35 \):
\[ a_4 = 3 \times 35 + 2 = 105 + 2 = 107 \]

### Step 5: Calculate \( a_5 \)
Using the formula for \( n = 5 \):
\[ a_5 = 3a_{5-1} + 2 = 3a_4 + 2 \]
Substituting \( a_4 = 107 \):
\[ a_5 = 3 \times 107 + 2 = 321 + 2 = 323 \]

### Summary of the First Five Terms
The first five terms of the sequence are:
\[ a_1 = 3, \quad a_2 = 11, \quad a_3 = 35, \quad a_4 = 107, \quad a_5 = 323 \]

### Final Answer
The first five terms of the sequence are:
\[ 3, 11, 35, 107, 323 \]