# Using induction to prove $f(n)=2n-3$ if $n\lt4$, $f(n)=2n-4$ if $n\ge4$ [closed]

I have a problem, for which the solution(by looking at the pattern) I found is

$$f(n)=\begin{cases}2n-3,\text{if }n<4\\2n-4,\text{if }n\ge 4\;.\end{cases}$$

I want to prove it inductively, I'm confused as to where do I begin? I know the basic induction proving it for $n=1$, assuming it true for $n=1,\dots,n-1$, and then proving it for $n=n$.

The problem is here $f(n)$ for $n=1$ is different and for $n-1$ (if greater than $3$ it will be different). If someone can guide me... how do I go on proving this thing.

Update: I have heard that induction works because we prove it for n=1, assume it true for n=1... n-1 and then prove it for n=n... which works because proving it for n=n is similar to n=1 and it again proves for n=(n+1)... (2n-1), we keep on going like that.

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You would prove cases $n=1,2$, and $3$ separately and then prove the case $n\ge 4$ by induction starting at $n=4$ as your base case. A proof by induction need not start at $1$. – Brian M. Scott Apr 12 '12 at 0:44
I thought that but I heard that when we prove it for n.. it basically rounds back to n=1(intuitively) and since we have already proved n=1, its proved for all the numbers. – questions Apr 12 '12 at 0:46
That dispayed version of $f(n)$ is exactly equivalent to your original two-line statement. I don’t understand your other comment; perhaps you should add the original problem to your question. – Brian M. Scott Apr 12 '12 at 0:49
Due to some reason, the formatting is not showing up in firefox.. and I have updated the question. – questions Apr 12 '12 at 0:50
@questions: If you really insist on starting at $1$ (and you definitely should not), let $g(n)=f(n+3)$. You can show by induction on $m$ that $g(m)=2(m+3)-4$ for all $m\ge 1$. But to repeat what others have said, use as base case $n=4$. – André Nicolas Apr 12 '12 at 3:41