Let a sequence `x_(1),x_(2),x_(3),…` of complex numbers be defined by `x_(1)=0, x_(n+1)=x_(n)^(2)-i` for all ` n gt 1`, where `i^(2)=-1`. Find the distance of `x_(2000)` from `x_(1997)` in the complex plane.

Answer

To find the distance between \( x_{2000} \) and \( x_{1997} \) in the complex plane, we start by analyzing the sequence defined by \( x_1 = 0 \) and \( x_{n+1} = x_n^2 - i \).

### Step 1: Calculate \( x_2 \)
\[ x_2 = x_1^2 - i = 0^2 - i = -i \]

**Hint:** Substitute \( x_1 \) into the recurrence relation to find \( x_2 \).

### Step 2: Calculate \( x_3 \)
\[ x_3 = x_2^2 - i = (-i)^2 - i = -1 - i \]

**Hint:** Remember that \( (-i)^2 = -1 \).

### Step 3: Calculate \( x_4 \)
\[ x_4 = x_3^2 - i = (-1 - i)^2 - i \]
Calculating \( (-1 - i)^2 \):
\[ (-1 - i)^2 = 1 + 2i + (-1) = 2i \]
Thus,
\[ x_4 = 2i - i = i \]

**Hint:** Use the formula \( (a + b)^2 = a^2 + 2ab + b^2 \) to expand \( x_3^2 \).

### Step 4: Calculate \( x_5 \)
\[ x_5 = x_4^2 - i = i^2 - i = -1 - i \]

**Hint:** Recall that \( i^2 = -1 \).

### Step 5: Calculate \( x_6 \)
\[ x_6 = x_5^2 - i = (-1 - i)^2 - i = 2i - i = i \]

**Hint:** Notice that \( x_5 \) is the same as \( x_3 \).

### Step 6: Identify the pattern
From the calculations:
- \( x_2 = -i \)
- \( x_3 = -1 - i \)
- \( x_4 = i \)
- \( x_5 = -1 - i \) (same as \( x_3 \))
- \( x_6 = i \) (same as \( x_4 \))

Thus, we see that:
- For even \( n \), \( x_{2n} = i \)
- For odd \( n \), \( x_{2n-1} = -1 - i \)

### Step 7: Find \( x_{2000} \) and \( x_{1997} \)
- \( x_{2000} = i \)
- \( x_{1997} = -1 - i \)

### Step 8: Calculate the distance
The distance \( d \) between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) in the complex plane is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Here, \( x_{2000} = 0 + 1i \) and \( x_{1997} = -1 - 1i \).

Calculating the differences:
- \( x_2 - x_1 = 0 - (-1) = 1 \)
- \( y_2 - y_1 = 1 - (-1) = 2 \)

Now substituting into the distance formula:
\[ d = \sqrt{(1)^2 + (2)^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Final Answer
The distance between \( x_{2000} \) and \( x_{1997} \) is \( \sqrt{5} \).

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