If f: R R is defined as f(x+y)=f(x)+f(y) , x, y R and f( l )=5 , then find the value of the following _ r=1 ^n f(r) is equal to

Step-by-step solution

Given f: R R is defined as f(x+y)=f(x)+f(y), x, y R and f(1)=5 To Find _ r=1 ^n f(r)= ? Since, f(x+y)=f(x)+f(y) Let x=y=1 aligned f(1+1) & =f( l )+f( l )=5+5 f(2) & =10 aligned Again in Eq. (i), put x=2, y=1 aligned f(2+1) & =f(2)+f(1) f(3) & =10+5 f(3) & =15 aligned Similarly, we can find f(4)=15+5=20 aligned f(5) & =25, f(n)=5 n _ r=1 ^n f(r) & =f(1)+f(2)+ +f(n) & =5+10+15+ +5 n aligned which forms an AP of n -terms with first term (a) =5 and common difference (d)=5 aligned & = n 2 [2(5)+(n-1) 5] & = n 2 [10+5 n-