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.
Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.
- Proizvolov's Identity
- Significance of Pascal’s Identity
- Program for Identity Matrix
- Brahmagupta Fibonacci Identity
- Euler's Four Square Identity
- Factorial of Large numbers using Logarithmic identity
- Generate a String of having N*N distinct non-palindromic Substrings
- Count of all possible Paths in a Tree such that Node X does not appear before Node Y
- Count of Array elements greater than all elements on its left and next K elements on its right
- Count of Array elements greater than or equal to twice the Median of K trailing Array elements
- Largest possible value of M not exceeding N having equal Bitwise OR and XOR between them
- Minimum number of operations required to reduce N to 0
- Smallest positive integer X satisfying the given equation
- Generate all possible permutations of a Number divisible by N