<s>[INST] Recursion Question - Trying to understand the concept Just trying to grasp this concept and was hoping someone could help me a bit.  I am taking a discrete math class.  Can someone please explain this equation to me a bit?
$f(0) = 3$
$f(n+1) = 2f(n) + 3$
$f(1) = 2f(0) + 3 = 2 \cdot 3 + 3 = 9$
$f(2) = 2f(1) + 3 = 2 \cdot 9 + 3 = 21$
$f(3) = 2f(2) + 3 = 2 \cdot 21 + 3 = 45$
$f(4) = 2f(3) + 3 = 2 \cdot 45 + 3 = 93$
I do not see how they get the numbers to the right of the equals sign.  Please someone show me how $f(2) = 2f(1) + 3 = 2 \cdot 9 + 3$. I see they get "$2\cdot$" because of $2f$ but how and where does the $9$ come from?  I also see why the $+3$ at the end of each equation but how and where does that number in the middle come from?
 [/INST] The function $f$ is given by a starting condition $f(0)=3$, and by the condition $f(n+1)=2\cdot f(n)+3$. This means that we calculate the value of $f(2)$ from the value of $f(1)$, which itself is calculated from $f(0)$ -- which was given to us.
$$f(1)=9\implies f(2)=2\cdot f(1)+3=2\cdot 9+3$$
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