Let `f:R rarr R ` be a function which satisfies `f(x+y)=f(x)+f(y) AA x, y in R`. If `f(1)=2` and `g(n)=sum_(k-1)^((n-1))f(k),n in N` then the value of n , for which `g(n)=20`, is :

Options

A. 9
B. 5
C. 4
D. 20

Answer

To solve the problem step by step, we will follow the given conditions and derive the necessary equations.

### Step 1: Understand the properties of the function \( f \)
The function \( f \) satisfies the property:
\[ f(x+y) = f(x) + f(y) \quad \forall x, y \in \mathbb{R} \]
This property indicates that \( f \) is a linear function. We also know that \( f(1) = 2 \).

### Step 2: Determine the form of \( f(x) \)
Since \( f \) is a linear function, we can express it in the form:
\[ f(x) = cx \]
for some constant \( c \). Given that \( f(1) = 2 \), we can substitute \( x = 1 \):
\[ f(1) = c \cdot 1 = c = 2 \]
Thus, we have:
\[ f(x) = 2x \]

### Step 3: Define the function \( g(n) \)
The function \( g(n) \) is defined as:
\[ g(n) = \sum_{k=1}^{n-1} f(k) \]
Substituting the expression for \( f(k) \):
\[ g(n) = \sum_{k=1}^{n-1} 2k = 2 \sum_{k=1}^{n-1} k \]

### Step 4: Calculate the sum \( \sum_{k=1}^{n-1} k \)
The sum of the first \( n-1 \) natural numbers is given by the formula:
\[ \sum_{k=1}^{n-1} k = \frac{(n-1)n}{2} \]
Thus, we can rewrite \( g(n) \):
\[ g(n) = 2 \cdot \frac{(n-1)n}{2} = (n-1)n \]

### Step 5: Set up the equation to find \( n \)
We need to find \( n \) such that:
\[ g(n) = 20 \]
Substituting our expression for \( g(n) \):
\[ (n-1)n = 20 \]
This simplifies to:
\[ n^2 - n - 20 = 0 \]

### Step 6: Solve the quadratic equation
We can solve the quadratic equation using the factorization method:
\[ n^2 - 5n + 4n - 20 = 0 \]
Factoring gives:
\[ (n - 5)(n + 4) = 0 \]
Thus, we have two solutions:
\[ n - 5 = 0 \quad \Rightarrow \quad n = 5 \]
\[ n + 4 = 0 \quad \Rightarrow \quad n = -4 \]

### Step 7: Determine the valid solution
Since \( n \) must be a natural number, we discard \( n = -4 \). Therefore, the only valid solution is:
\[ n = 5 \]

### Final Answer
The value of \( n \) for which \( g(n) = 20 \) is:
\[ \boxed{5} \]