Understanding how recursive functions work

Question

As the title explains I have a very fundamental programming question which I have just not been able to grok yet. Filtering out all of the (extremely clever) \"In order to

Answer 1

Let me tell you with an example of Fibonacci series, Fibonacci is

t(n) = t(n - 1) + n;

if n = 0 then 1

so let see how recursion works, I just replace n in t(n) with n-1 and so on. it looks:

t(n-1) = t(n - 2) + n+1;

t(n-1) = t(n - 3) + n+1 + n;

t(n-1) = t(n - 4) + n+1 + n+2 + n;

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t(n) = t(n-k)+ ... + (n-k-3) + (n-k-2)+ (n-k-1)+ n ;

we know if t(0)=(n-k) equals to 1 then n-k=0 so n=k we replace k with n:

t(n) = t(n-n)+ ... + (n-n+3) + (n-n+2)+ (n-n+1)+ n ;

if we omit n-n then:

t(n)= t(0)+ ... + 3+2+1+(n-1)+n;

so 3+2+1+(n-1)+n is natural number. it calculates as Σ3+2+1+(n-1)+n = n(n+1)/2 => n²+n/2

the result for fib is : O(1 + n²) = O(n²)

This the best way to understand recursive relation