If `f:RtoR` satisfies f(x+y)=f(x)+f(y) for all x,y `in` R and f(1)=7, then `sum_(r=1)^(n) f(r)`, is

Options

A. `3^(m)-1`
B. `3^(m)`
C. `3^(m-1)`
D. none of these

Explanation

To solve the problem step by step, we start with the functional equation given and the value of the function at a specific point.

### Step 1: Understanding the Functional Equation
We are given that \( f: \mathbb{R} \to \mathbb{R} \) satisfies the equation:
\[ f(x+y) = f(x) + f(y) \quad \text{for all } x, y \in \mathbb{R} \]
This is known as Cauchy's functional equation.

### Step 2: Finding \( f(2) \)
We know \( f(1) = 7 \). Let's find \( f(2) \) by substituting \( x = 1 \) and \( y = 1 \):
\[ f(1 + 1) = f(1) + f(1) \implies f(2) = 2f(1) = 2 \times 7 = 14 \]

### Step 3: Finding \( f(3) \)
Next, we can find \( f(3) \) by substituting \( x = 2 \) and \( y = 1 \):
\[ f(2 + 1) = f(2) + f(1) \implies f(3) = f(2) + f(1) = 14 + 7 = 21 \]

### Step 4: Finding \( f(4) \)
Now, let's find \( f(4) \) by substituting \( x = 3 \) and \( y = 1 \):
\[ f(3 + 1) = f(3) + f(1) \implies f(4) = f(3) + f(1) = 21 + 7 = 28 \]

### Step 5: Generalizing \( f(n) \)
From the pattern observed, we can generalize that:
\[ f(n) = n \cdot f(1) = n \cdot 7 = 7n \]
for any natural number \( n \).

### Step 6: Finding the Summation
We need to calculate:
\[ \sum_{r=1}^{n} f(r) \]
Substituting our expression for \( f(r) \):
\[ \sum_{r=1}^{n} f(r) = \sum_{r=1}^{n} 7r = 7 \sum_{r=1}^{n} r \]
The sum of the first \( n \) natural numbers is given by:
\[ \sum_{r=1}^{n} r = \frac{n(n + 1)}{2} \]
Thus,
\[ \sum_{r=1}^{n} f(r) = 7 \cdot \frac{n(n + 1)}{2} = \frac{7n(n + 1)}{2} \]

### Final Result
The final result is:
\[ \sum_{r=1}^{n} f(r) = \frac{7n(n + 1)}{2} \]