the expressions are : f(1) = (a^0) f(2) = (a+1) f(3) = (a^2+a+1)
and the answer is f(n)= (a^n-1) /a-1, it is the formula for the sum of the geometric series right ?
I have tried to find the formula but I stuck in this : f(1)= a^(n-1) f(2)= a^(n-1)+a^(n-2) f(3)= a^(n-1)+a^(n-2)+a^(n-3) how can I make the 3 equations below to find that the formula is the sum of geometric series ,,
Assuming that by f^1 you mean $f(1)$ and so on there is no way to find that $f(n)$ is the sum of the geometric series aside from recognizing the pattern. Nothing in the data you are given tells you what $f(4)$ is. The geometric series is (to my eye) the simplest explanation for the first three terms and problems where you are given the first few terms of a series or sequence are intended to be solved by finding the simplest pattern (to the setter's eye) and extending it.
Whenever somebody posts a sequence or series problem there are those who complain (I would say correctly but pedantically) that you can extend the sequence or series any way you want so there is no solution. I believe the setter owes us enough information to make one solution stand out as the simplest and it can be a fun problem to find it. Here I would go with the geometric series as you suggest.