Question

Let \(f : \mathbb{R} \to \mathbb{R}\) be a function which satisfies \(f(x+y) = f(x) + f(y)\ \forall\ x, y \in \mathbb{R}\). If \(f(1) = 2\) and \(g(n) = \sum\limits_{k=1}^{(n-1)} f(k)\), \(n \in \mathbb{N}\), then the value of n, for which \(g(n) = 20\), is :

• A. 20
• B. 9
• C. 5
• D. 4

Hint:

Additivity with \(f(1)=2\) gives \(f(k)=2k\) for natural k, so \(g(n)=n(n-1)\).

Answer 1

The Correct Option is C

\(f(x+1)=f(x)+f(1)=f(x)+2\), and \(f(0)=0\), so by induction \(f(k)=2k\) for \(k\in\mathbb N\).
\(g(n)=\sum_{k=1}^{n-1}2k = 2\cdot\frac{(n-1)n}{2} = n(n-1)\).
\(n(n-1)=20\Rightarrow n^2-n-20=0\Rightarrow (n-5)(n+4)=0\Rightarrow n=5\) (n is natural).
Answer: (3) 5