If a= 297, b= 298, c= 299 and find `a^(2) + b^(2) + c^(2) - ab - bc - ca` ?

Answer

To solve the expression \( a^2 + b^2 + c^2 - ab - bc - ca \) given \( a = 297 \), \( b = 298 \), and \( c = 299 \), we can use a formula that simplifies the calculation. The formula states:

\[ a^2 + b^2 + c^2 - ab - bc - ca = \frac{1}{2} \left( (a-b)^2 + (b-c)^2 + (c-a)^2 \right) \]

### Step-by-step Solution:

1. **Identify the values of a, b, and c:**
\[ a = 297, \quad b = 298, \quad c = 299 \]

2. **Calculate \( (a-b)^2 \):**
\[ a - b = 297 - 298 = -1 \quad \Rightarrow \quad (a-b)^2 = (-1)^2 = 1 \]

3. **Calculate \( (b-c)^2 \):**
\[ b - c = 298 - 299 = -1 \quad \Rightarrow \quad (b-c)^2 = (-1)^2 = 1 \]

4. **Calculate \( (c-a)^2 \):**
\[ c - a = 299 - 297 = 2 \quad \Rightarrow \quad (c-a)^2 = (2)^2 = 4 \]

5. **Sum the squares:**
\[ (a-b)^2 + (b-c)^2 + (c-a)^2 = 1 + 1 + 4 = 6 \]

6. **Multiply by \(\frac{1}{2}\):**
\[ \frac{1}{2} \times 6 = 3 \]

### Final Result:
Thus, the value of \( a^2 + b^2 + c^2 - ab - bc - ca \) is:
\[ \boxed{3} \]