let f(x) be continuous function such that f(0)=1 and f(x)-f(x/7)= x/7 then f(42)

Question

Solve 

 

If f=1 then f=42

Answer 1

Then f(x) = f(x/7) + x/7 <--- recursive definition of the function

f(42) = f(42/7) + 42/7

= f(6) + 6

= f(6/7) + 6/7 + 6

= f(6/49) + 6/49 + 6/7 + 6

... etc....

This is a geometric series: 6 ( 1/7)^n which converges to 6 [ 1 - 1/7]^(-1)

= 6(6/7)^-1 = 6*7/6 = 7