Let f : R [ ] R be a function which satisfies f(x + y) = f(x) + f(y) [ ] x, y [ ] R.

Question

Let f : R $$ \to $$ R be a function which satisfies

f(x + y) = f(x) + f(y) $$\forall $$ x, y $$ \in $$ R. If f(1) = 2 and

g(n) = $$\sum\limits_{k = 1}^{\left( {n - 1} \right)} {f\left( k \right)} $$, n $$ \in $$ N then the value of n, for
which g(n) = 20, is :

Accepted Answer

Given f(1) = 2 ;

f(x + y) = f(x) + f(y)

When x = y = 1 $$ \Rightarrow $$
f(2) = 2 + 2 = 4

When x = 2, y = 1 $$ \Rightarrow $$ f(3) = 4 + 2 = 6

g(n) = $$\sum\limits_{k = 1}^{\left( {n - 1} \right)} {f\left( k \right)} $$

= f(1) + f(2) +.........+ f(n - 1)

= 2 + 4 + 6 + ......+ 2(n - 1)

= 2 $$ \times $$ $$\frac{\left( n-1\right)  \left( n\right)  }{2} $$

= n2 - n

Given g(n) = 20

$$ \Rightarrow $$ n2-- n = 20

$$ \Rightarrow $$ n2 -- n -- 20 = 0

$$ \Rightarrow $$ n = 5

Solutions · 1