Question

Let \( f: \mathbb{R} \to \mathbb{R} \) be a function defined by \( f(x) = \left( 2 + 3a \right)x^2 + \left( \frac{a+2}{a-1} \right)x + b, a \neq 1 \). If \[ f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy, \] then the value of \( 28 \sum_{i=1}^5 f(i) \) is:

• A. 715
• B. 735
• C. 545
• D. 675

Hint:

Be mindful of the properties of functions when solving for unknowns in functional equations.

Answer 1

The Correct Option is B

We are given the functional equation \( f(x + y) = f(x) + f(y) + 1 - \frac{2}{7}xy \) and the form of \( f(x) \). 
Step 1: First, solve for the values of \( a \) and \( b \) by substituting \( x = y = 0 \) into the functional equation. This simplifies the equation. 
Step 2: For \( f(x) \), substitute the given expression for \( f(x) \) and use the relation from step 1 to find \( a \) and \( b \). 
Step 3: Once we have \( a \) and \( b \), calculate \( f(x) \) for \( x = 1, 2, 3, 4, 5 \). 
Step 4: Now calculate \( 28 \sum_{i=1}^5 f(i) \) by plugging the values of \( f(i) \) into the summation. 

Final Conclusion: The value of \( 28 \sum_{i=1}^5 f(i) \) is 735, which is Option 2.