Some sequences are defined as follows. Find their first four terms :
(i) `a_(1)=a_(2)=2, a_(n)=a_(n-1)-1, n gt 2 " " (ii) a_(1)=3, a_(n)=3a_(n-1), n gt 1`

Answer

To find the first four terms of the given sequences, we will solve each part step by step.

### Part (i)
The sequence is defined as:
- \( a_1 = a_2 = 2 \)
- \( a_n = a_{n-1} - 1 \) for \( n > 2 \)

**Step 1:** Identify the first two terms.
- \( a_1 = 2 \)
- \( a_2 = 2 \)

**Step 2:** Calculate the third term \( a_3 \).
- Using the formula:
\[ a_3 = a_{2} - 1 = 2 - 1 = 1 \]

**Step 3:** Calculate the fourth term \( a_4 \).
- Using the formula:
\[ a_4 = a_{3} - 1 = 1 - 1 = 0 \]

**Conclusion for Part (i):** The first four terms are:
\[ 2, 2, 1, 0 \]

### Part (ii)
The sequence is defined as:
- \( a_1 = 3 \)
- \( a_n = 3 \cdot a_{n-1} \) for \( n > 1 \)

**Step 1:** Identify the first term.
- \( a_1 = 3 \)

**Step 2:** Calculate the second term \( a_2 \).
- Using the formula:
\[ a_2 = 3 \cdot a_{1} = 3 \cdot 3 = 9 \]

**Step 3:** Calculate the third term \( a_3 \).
- Using the formula:
\[ a_3 = 3 \cdot a_{2} = 3 \cdot 9 = 27 \]

**Step 4:** Calculate the fourth term \( a_4 \).
- Using the formula:
\[ a_4 = 3 \cdot a_{3} = 3 \cdot 27 = 81 \]

**Conclusion for Part (ii):** The first four terms are:
\[ 3, 9, 27, 81 \]

### Final Answer
- For Part (i): The first four terms are \( 2, 2, 1, 0 \).
- For Part (ii): The first four terms are \( 3, 9, 27, 81 \).

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