Answer
See the explanation
Work Step by Step
To calculate the Fibonacci series of a given number efficiently, you can use either an iterative or recursive approach. Both approaches have their pros and cons in terms of efficiency and space complexity.
Here's a simple iterative algorithm to calculate the Fibonacci series:
```python
def fibonacci(n):
if n <= 1:
return n
else:
fib_list = [0, 1]
for i in range(2, n+1):
fib_list.append(fib_list[i-1] + fib_list[i-2])
return fib_list[n]
# Example usage:
result = fibonacci(10)
print(result) # Output: 55
```
This iterative approach has a time complexity of O(n) and a space complexity of O(n) since it stores the entire Fibonacci series in a list.
Alternatively, you can use a more space-efficient iterative approach that only keeps track of the last two Fibonacci numbers:
```python
def fibonacci(n):
if n <= 1:
return n
else:
a, b = 0, 1
for _ in range(2, n+1):
a, b = b, a + b
return b
# Example usage:
result = fibonacci(10)
print(result) # Output: 55
```
This approach still has a time complexity of O(n) but reduces the space complexity to O(1) since it only stores two variables.
Both of these solutions are polynomial in time complexity, specifically linear (O(n)), making them efficient for calculating Fibonacci numbers.
