Function? How?

We have $\large f(2008)$ =3012.

The question asks for $\large f(2009)$ = $\large f(2008 + 1)$ = $\large f(2008) + f(1)$ = $\large 3012 + f(1)$ . So, we just have to find $\large f(1)$ & we are done!

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Now lets plug in values and try to observe a pattern.

$\large f(0)$ = $\large 2f(0)$ . Clearly then $\large f(0)=0$

$\large f(1)$ = $\large f(1)$ + $\large f(0)$ = $\large f(1)$ ( unnecessary step just to add to the pattern :D)

$\large f(2)$ = $\large f(1)$ + $\large f(1)$ = $\large 2f(1)$

$\large f(3)$ = $\large f(2)$ + $\large f(1)$ = $\large 3f(1)$

$\large f(4)$ = $\large f(3)$ + $\large f(1)$ = $\large f(2)$ + $\large f(2)$ = $\large 4f(4)$

.....

Observe that $\large f(n)$ = $\large nf(1)$ .

Then, $\large f(2008)$ = $\large 2008f(1)$ ;

Hence, $\large f(1)$ = 1.5.