Given a number N, the task is to evaluate below expression. Expected time complexity is O(1).
f(n-1)*f(n+1) - f(n)*f(n)
Where f(n) is the n-th Fibonacci number with n >= 1. First few Fibonacci numbers are 0, 1, 1, 2, 3, 5, 8, 13, ………..i.e. (considering 0 as 0th Fibonacci number)
Input : n = 5 Output : -1 f(5-1=4) = 3 f(5+1=6) = 8 f(5)*f(5)= 5*5 = 25 f(4)*f(6)- f(5)*f(5)= 24-25= -1
Although the task is simple i.e. find n-1th, nth and (n+1)-th Fibonacci numbers. Evaluate the expression and display the result. But this can be done in O(1) time using Cassini’s Identity which states that:
f(n-1)*f(n+1) - f(n*n) = (-1)^n
So, we don’t need to calculate any Fibonacci term,the only thing is to check whether n is even or odd.
This article is contributed by Sahil Chhabra. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to email@example.com. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.
- Proizvolov's Identity
- Significance of Pascal’s Identity
- Program for Identity Matrix
- Euler's Four Square Identity
- Brahmagupta Fibonacci Identity
- Queries to check whether bitwise AND of a subarray is even or odd
- Find distinct integers for a triplet with given product
- Making three numbers equal with the given operations
- Number of valid indices in the permutation of first N natural numbers
- Check if Sum and XOR of all elements of array is equal
- Arrange numbers to form a valid sequence
- Count the subarray with sum strictly greater than the sum of remaining elements
- Find smallest number with given number of digits and sum of digits under given constraints
- Count of odd length contiguous Palindromic sequences in a Matrix