Key Concepts and Formulas

• Cauchy's Functional Equation: The equation $f(x+y) = f(x) + f(y)$ for all $x, y$ in the domain of $f$. For integer inputs, this implies $f(n) = nf(1)$.
• Sum of First $n$ Natural Numbers: The formula for the sum of an arithmetic series: $\sum_{r=1}^n r = \frac{n(n+1)}{2}$.
• Function Properties: Understanding how to derive properties of a function from a given functional equation and initial conditions.

Step-by-Step Solution

Step 1: Analyze the Functional Equation and Derive $f(n)$ for Integers We are given the functional equation $f(x+y) = f(x) + f(y)$ for all $x, y \in R$. This is known as Cauchy's functional equation. Our goal is to find $\sum_{r=1}^n f(r)$, which involves integer values of $r$. We can deduce the form of $f(n)$ for integers from the given equation.

First, let $x=0$ and $y=0$: $f(0+0) = f(0) + f(0)$ $f(0) = 2f(0)$ Subtracting $f(0)$ from both sides gives $f(0) = 0$.

Next, let's find $f(n)$ for positive integers $n$. For $n=1$, we are given $f(1) = 7$. For $n=2$: $f(2) = f(1+1) = f(1) + f(1) = 2f(1)$ For $n=3$: $f(3) = f(2+1) = f(2) + f(1) = 2f(1) + f(1) = 3f(1)$ By induction, we can show that for any positive integer $n$, $f(n) = nf(1)$.

Now, let's consider negative integers. Let $y = -x$: $f(x+(-x)) = f(x) + f(-x)$ $f(0) = f(x) + f(-x)$ Since $f(0)=0$, we have $0 = f(x) + f(-x)$, which implies $f(-x) = -f(x)$. For a negative integer $-n$ (where $n > 0$): $f(-n) = -f(n) = -(nf(1)) = (-n)f(1)$ Thus, the relation $f(n) = nf(1)$ holds for all integers $n$ (positive, negative, and zero).

Why this step is taken: The problem requires summing $f(r)$ for integer values of $r$. Establishing the general form of $f(n)$ for integers simplifies the subsequent calculation of the sum.

Step 2: Use the Given Condition $f(1) = 7$ to Determine the Specific Function We are given that $f(1) = 7$. Substituting this into our derived relation $f(n) = nf(1)$: $f(n) = n \cdot 7$ $f(n) = 7n$ So, for any integer $n$, the function evaluates to $7n$.

Why this step is taken: This step uses the specific information provided in the problem to find the exact form of the function for integer inputs, which is necessary to evaluate the summation.

Step 3: Evaluate the Summation $\sum_{r=1}^n f(r)$ We need to calculate $\sum_{r=1}^n f(r)$. Using our finding from Step 2, $f(r) = 7r$: $\sum_{r=1}^n f(r) = \sum_{r=1}^n (7r)$ We can factor out the constant $7$ from the summation: $= 7 \sum_{r=1}^n r$ The summation $\sum_{r=1}^n r$ is the sum of the first $n$ natural numbers, for which the formula is $\frac{n(n+1)}{2}$. Substituting this formula: $= 7 \left( \frac{n(n+1)}{2} \right)$ $= \frac{7n(n+1)}{2}$

Why this step is taken: This is the final calculation step, where we apply a standard summation formula to compute the required sum, using the specific form of $f(r)$ derived earlier.

Common Mistakes & Tips

• Assuming $f(x)=cx$ for all $x \in R$ without justification: While $f(x)=cx$ is often the intended solution for $f:R \to R$ in such problems, its rigorous proof for all real numbers requires additional conditions like continuity. However, for integer inputs, $f(n)=nf(1)$ is directly derivable from the functional equation, making the solution valid.
• Forgetting Summation Formulas: Be sure to memorize common summation formulas like the sum of the first $n$ natural numbers.
• Algebraic Errors: Double-check all algebraic manipulations, especially when factoring out constants or applying formulas.

Summary

The problem involves a function satisfying Cauchy's functional equation, $f(x+y) = f(x) + f(y)$. For integer inputs, this implies $f(n) = nf(1)$. Given $f(1) = 7$, we find that $f(n) = 7n$. The required summation $\sum_{r=1}^n f(r)$ becomes $\sum_{r=1}^n 7r$, which simplifies to $7 \sum_{r=1}^n r$. Using the formula for the sum of the first $n$ natural numbers, we arrive at $\frac{7n(n+1)}{2}$.

The final answer is $\boxed{\frac{7n\left( {n + 1} \right)}{2}}$, which corresponds to option (A).