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The sum of first N odd index Fibonacci numbers refers to the total obtained by adding up the Fibonacci numbers that appear at positions with odd indices (1st, 3rd, 5th, etc.) in the Fibonacci sequence.
The calculator uses the formula:
Where:
Explanation: This formula provides a direct way to calculate the sum of Fibonacci numbers at odd indices without having to sum them individually.
Details: The Fibonacci sequence appears in various natural phenomena and mathematical applications. Understanding properties of Fibonacci sums helps in combinatorial mathematics, computer algorithms, and number theory.
Tips: Enter a positive integer N to calculate the sum of first N Fibonacci numbers at odd indices. The calculator will compute the result using the direct formula.
Q1: What are the first few odd index Fibonacci numbers?
A: The Fibonacci sequence starts: 0, 1, 1, 2, 3, 5, 8, 13... The odd index terms (1st, 3rd, 5th, etc.) are: 1, 1, 3, 8, 21...
Q2: Why does this formula work?
A: The formula works due to mathematical properties and recurrence relations in the Fibonacci sequence that establish this relationship between odd-index sums and the 2n-th term.
Q3: What is the time complexity of this calculation?
A: Using the direct formula approach has constant time complexity O(1) for the sum calculation, though computing F₂n may have higher complexity depending on the method used.
Q4: Are there similar formulas for even indices?
A: Yes, there are similar formulas for sums of Fibonacci numbers at even indices, which also relate to other terms in the Fibonacci sequence.
Q5: What are some applications of Fibonacci sums?
A: Fibonacci sums find applications in algorithm analysis, combinatorial mathematics, financial modeling, and various areas of computer science and mathematics.