We have a sequence `a_1, a_2, ....oo` where `a_1 = 4444, a_2 = - 1234` and `a_(k + 2) = a_(k+1)-a_k, k gt= 1.` The sum of first 404 terms is

Answer

To find the sum of the first 404 terms of the sequence defined by \( a_1 = 4444 \), \( a_2 = -1234 \), and the recurrence relation \( a_{k+2} = a_{k+1} - a_k \) for \( k \geq 1 \), we will proceed step by step.

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

1. **Given:**
- \( a_1 = 4444 \)
- \( a_2 = -1234 \)

2. **Calculate \( a_3 \):**
\[ a_3 = a_2 - a_1 = -1234 - 4444 = -5678 \]

3. **Calculate \( a_4 \):**
\[ a_4 = a_3 - a_2 = -5678 - (-1234) = -5678 + 1234 = -4444 \]

4. **Calculate \( a_5 \):**
\[ a_5 = a_4 - a_3 = -4444 - (-5678) = -4444 + 5678 = 1234 \]

5. **Calculate \( a_6 \):**
\[ a_6 = a_5 - a_4 = 1234 - (-4444) = 1234 + 4444 = 5678 \]

6. **Calculate \( a_7 \):**
\[ a_7 = a_6 - a_5 = 5678 - 1234 = 4444 \]

7. **Calculate \( a_8 \):**
\[ a_8 = a_7 - a_6 = 4444 - 5678 = -1234 \]

8. **Calculate \( a_9 \):**
\[ a_9 = a_8 - a_7 = -1234 - 4444 = -5678 \]

9. **Calculate \( a_{10} \):**
\[ a_{10} = a_9 - a_8 = -5678 - (-1234) = -5678 + 1234 = -4444 \]

From these calculations, we can observe a repeating pattern in the sequence:
- \( a_1 = 4444 \)
- \( a_2 = -1234 \)
- \( a_3 = -5678 \)
- \( a_4 = -4444 \)
- \( a_5 = 1234 \)
- \( a_6 = 5678 \)

### Step 2: Identify the repeating cycle

The sequence shows a periodicity of 6:
- \( a_1, a_2, a_3, a_4, a_5, a_6 \) repeat as \( 4444, -1234, -5678, -4444, 1234, 5678 \).

### Step 3: Calculate the sum of one complete cycle

The sum of one complete cycle (6 terms) is:
\[ S = 4444 + (-1234) + (-5678) + (-4444) + 1234 + 5678 \]
Calculating this step-by-step:
- \( 4444 - 1234 = 3210 \)
- \( 3210 - 5678 = -2468 \)
- \( -2468 - 4444 = -6912 \)
- \( -6912 + 1234 = -6678 \)
- \( -6678 + 5678 = -1000 \)

Thus, the sum of one complete cycle is:
\[ S = -1000 \]

### Step 4: Calculate the total number of complete cycles in 404 terms

To find how many complete cycles fit into 404 terms:
\[ \text{Number of complete cycles} = \left\lfloor \frac{404}{6} \right\rfloor = 67 \]
The number of terms in these complete cycles:
\[ 67 \times 6 = 402 \]

### Step 5: Calculate the sum of the complete cycles

The sum of the first 402 terms (67 complete cycles) is:
\[ \text{Sum of 402 terms} = 67 \times (-1000) = -67000 \]

### Step 6: Add the remaining terms

The remaining terms are \( a_{403} \) and \( a_{404} \):
- \( a_{403} = a_1 = 4444 \)
- \( a_{404} = a_2 = -1234 \)

Calculating the sum of these remaining terms:
\[ \text{Sum of remaining terms} = 4444 + (-1234) = 3210 \]

### Step 7: Calculate the final sum

The total sum of the first 404 terms is:
\[ \text{Total Sum} = -67000 + 3210 = -63790 \]

### Final Answer
The sum of the first 404 terms is:
\[ \boxed{-63790} \]