Fabonacci series : 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233，377，610，987，1597，2584...

rule：The third number is equal to the sum of the first two Numbers
I once ran into an interview question: implementing the Fibonacci function.

• Implement with Recursion

- (NSInteger)recursionFibonacciWithIndex:(NSInteger)index{
    while (index <= 1) {
        return 1;
    }
    return ([self recursionFibonacciWithIndex:index - 2] + [self recursionFibonacciWithIndex:index - 1]);
}

When index == 47 ,I found the program will run a long time.
This arithmetic's Time Complexity is high(nest passage I will introduce Time Complexity), so I use another arithmetic.

• Implement with For

- (NSInteger)fibonacciWithIndex:(NSInteger)index{
    NSInteger p = 1;
    NSInteger q = 1;
    if (index == 0 || index == 1) {
        return 1;
    }else{
        for (NSInteger i = 2; i <= index; i++) {
            NSInteger tmp = p;
            p = q;
            q = tmp + p;
        }
        return q;
    }
}

Time Complexity is O(n).

Although recusion looks sample, but sometimes it will result Time Complexity higher