Consider the following sequence `:a _(1) = a _(2) =1, a _(i) = 1 +` minimum ` ( a _(i-1) , a _(i-2))` for `I gt 2.` Then `a _( 2006)=`

Answer

To solve the problem, we need to find the value of \( a_{2006} \) in the given sequence defined as follows:

1. \( a_1 = a_2 = 1 \)
2. For \( i > 2 \), \( a_i = 1 + \min(a_{i-1}, a_{i-2}) \)

Let's calculate the terms step by step.

### Step 1: Calculate the first few terms of the sequence

- **For \( i = 3 \)**:
\[ a_3 = 1 + \min(a_2, a_1) = 1 + \min(1, 1) = 1 + 1 = 2 \]

- **For \( i = 4 \)**:
\[ a_4 = 1 + \min(a_3, a_2) = 1 + \min(2, 1) = 1 + 1 = 2 \]

- **For \( i = 5 \)**:
\[ a_5 = 1 + \min(a_4, a_3) = 1 + \min(2, 2) = 1 + 2 = 3 \]

- **For \( i = 6 \)**:
\[ a_6 = 1 + \min(a_5, a_4) = 1 + \min(3, 2) = 1 + 2 = 3 \]

- **For \( i = 7 \)**:
\[ a_7 = 1 + \min(a_6, a_5) = 1 + \min(3, 3) = 1 + 3 = 4 \]

- **For \( i = 8 \)**:
\[ a_8 = 1 + \min(a_7, a_6) = 1 + \min(4, 3) = 1 + 3 = 4 \]

- **For \( i = 9 \)**:
\[ a_9 = 1 + \min(a_8, a_7) = 1 + \min(4, 4) = 1 + 4 = 5 \]

- **For \( i = 10 \)**:
\[ a_{10} = 1 + \min(a_9, a_8) = 1 + \min(5, 4) = 1 + 4 = 5 \]

### Step 2: Identify the pattern

From the calculations, we can observe the following pattern:

- The sequence appears to be:
\[ 1, 1, 2, 2, 3, 3, 4, 4, 5, 5, \ldots \]

This suggests that the even-indexed terms are equal to their half index, and the odd-indexed terms are the same as the previous even-indexed term.

### Step 3: Generalize the formula

From the pattern, we can deduce:
- For even \( n \) (i.e., \( a_{2n} \)), the term is \( n \).
- For odd \( n \) (i.e., \( a_{2n-1} \)), the term is also \( n \).

### Step 4: Calculate \( a_{2006} \)

Since \( 2006 \) is even, we can use the formula:
\[ a_{2006} = a_{2 \times 1003} = 1003 \]

### Final Answer
Thus, the value of \( a_{2006} \) is \( \boxed{1003} \).