If f:[R [R is such that f(x+y)=f(x)+f(y) for all x, y [R . f( l )=7 and _ r=1 ^n f(r)=14112 , then n=

Step-by-step solution

We have, f(x+y)=f(x)+f(y) and f(1)=7 array rlrl & & f(1) & =7 & & & f(2)=f(1)+f(1) & & f(2)=7+7=14 array Similarly, aligned & f(x)=f(1)+f(2)=7+14=21 & f(4)=f(1)+f(3)=7+21=28 & f(n)=7 n aligned f(n)=7 n Now, _ r=1 ^n f(r)=14112 array lr & f(1)+f(2)+f(3)+ +f(n)=14112 & 7+14+21+ +7 n=14112 & 7(1+2+3+ +n)=14112 & 7 n(n+1) 2 =14112 & n(n+1)=4032 & n(n+1)=63 64 n=63 array