Question

For all real numbers \( x \) and \( y \), it is known that the real valued function \( f \) satisfies \( f(x) + f(y) = f(x + y) \). If \( f(1) = 7 \), then \( \sum_{r=1}^{100} f(r) \) is equal to:

• A. \( 7 \times 51 \times 102 \)
• B. \( 6 \times 50 \times 102 \)
• C. \( 7 \times 50 \times 102 \)
• D. \( 6 \times 25 \times 102 \)
• E. \( 7 \times 50 \times 101 \)

Hint:

Cauchy's equation implies linearity. If you know \( f(1) \), then \( f(n) = n \cdot f(1) \). The sum then becomes \( f(1) \cdot \frac{n(n+1)}{2} \).

Answer 1

Answer: (E)

Concept: The functional equation \( f(x) + f(y) = f(x + y) \) is Cauchy's functional equation. For real-valued functions, the general solution is of the form \( f(x) = cx \), where \( c \) is a constant. Once the function is identified, the summation involves calculating the sum of the first \( n \) natural numbers using the formula \( S_n = \frac{n(n + 1)}{2} \).

Step 1: Identify the constant \( c \) for the function \( f(x) \).
Given \( f(x) = cx \). We are told that \( f(1) = 7 \). Substituting \( x = 1 \): \[ c(1) = 7 \implies c = 7 \] Thus, the function is defined as \( f(x) = 7x \).

Step 2: Set up and simplify the summation.
The problem asks for \( \sum_{r=1}^{100} f(r) \): \[ \sum_{r=1}^{100} 7r = 7 \sum_{r=1}^{100} r \] Using the sum of the first \( n \) natural numbers formula for \( n = 100 \): \[ \sum_{r=1}^{100} r = \frac{100(100 + 1)}{2} = \frac{100 \times 101}{2} = 50 \times 101 \]

Step 3: Calculate the final value.
Multiply the sum by the constant 7: \[ \text{Sum} = 7 \times (50 \times 101) = 7 \times 50 \times 101 \] Comparing this to the options, it matches Option (E).