Question

F(x) satisfies f(x+y)=f(x)+f(y) for all x,y belongs to R , f(1)=5 then find
m
singma [f(n)]
n=1

Answer 1

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f(x) satisfies f(x+y) = f(x) + f(y) for all x, y belongs to R.

Consider :

$f(x) = kx$

It satisfies the given equation

⇒$f(x+y) = k(x+y) = kx + ky$

⇒$f(x) = kx$

⇒$f(y) = ky$

$So f(x) + f(y) = f(x+y)$

According to the question,

$f(1) = 5$

⇒$k(1) = 5$

⇒ $k = 5$

Therefore, The function$f(x) = 5x$

To find :

refer to attachment

$5\frac{m(m+1)}{2}$

$5m\frac{(m+1)}{2}$

Therefore, The required answer is 5m(m+1)/2.

$\fbox {5m\frac{(m+1)}{2}}$$\boxed{\ 5m\frac{(m+1)}{2}}$

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