Write the first five terms of each of the sequences in Questions 11 to 13 and obtain the corresponding series :
`a_(1)=3,a_(n)=3a_(n-1)+2," for all "ngt1`

Explanation

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

### Step-by-Step Solution:

1. **Find \( a_1 \)**:
\[ a_1 = 3 \]

2. **Find \( a_2 \)**:
\[ a_2 = 3a_1 + 2 = 3 \times 3 + 2 = 9 + 2 = 11 \]

3. **Find \( a_3 \)**:
\[ a_3 = 3a_2 + 2 = 3 \times 11 + 2 = 33 + 2 = 35 \]

4. **Find \( a_4 \)**:
\[ a_4 = 3a_3 + 2 = 3 \times 35 + 2 = 105 + 2 = 107 \]

5. **Find \( a_5 \)**:
\[ a_5 = 3a_4 + 2 = 3 \times 107 + 2 = 321 + 2 = 323 \]

### Summary of the First Five Terms:
The first five terms of the sequence are:
- \( a_1 = 3 \)
- \( a_2 = 11 \)
- \( a_3 = 35 \)
- \( a_4 = 107 \)
- \( a_5 = 323 \)

### Corresponding Series:
The corresponding series can be written as:
\[ S = a_1 + a_2 + a_3 + a_4 + a_5 = 3 + 11 + 35 + 107 + 323 \]

### Calculate the Sum of the Series:
\[ S = 3 + 11 + 35 + 107 + 323 = 479 \]

### Final Answer:
The first five terms of the sequence are \( 3, 11, 35, 107, 323 \) and the corresponding series sums up to \( 479 \).

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