1. What is Fibonacci Series?

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, starting from 0 and 1. The sequence goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, etc. This calculator computes the sum of the first n Fibonacci numbers.

2. How Does the Calculator Work?

The calculator uses the Fibonacci sum formula:

Where:

• \( F_{n+2} \) — The (n+2)th Fibonacci number
• \( n \) — Number of terms to sum

Explanation: The sum of the first n Fibonacci numbers is equal to the (n+2)th Fibonacci number minus 1. This elegant mathematical property makes calculation efficient.

3. Importance of Fibonacci Series

Details: The Fibonacci sequence appears in many areas of mathematics and nature. It's used in computer algorithms, financial markets analysis, and appears in biological settings like branching in trees and arrangement of leaves.

4. Using the Calculator

Tips: Enter the number of Fibonacci terms you want to sum (n ≥ 1). The calculator will return the sum of the first n Fibonacci numbers (F₁ + F₂ + ... + Fₙ).

5. Frequently Asked Questions (FAQ)

Q1: What is the sum of first 10 Fibonacci numbers?
A: The sum is 143 (1+1+2+3+5+8+13+21+34+55 = 143).

Q2: Why does the formula F(n+2)-1 work?
A: This comes from mathematical induction and the recursive definition of Fibonacci numbers. Each sum builds on the previous sums in a predictable way.

Q3: What's the largest n this calculator can handle?
A: The calculator can handle very large n values (up to PHP's integer limit), but extremely large values may cause performance issues.

Q4: Are there other ways to calculate the sum?
A: Yes, you could calculate each Fibonacci number sequentially and add them, but the formula method is more efficient.

Q5: Does this include F₀=0 in the sum?
A: No, this calculator uses the modern convention starting with F₁=1, F₂=1. Some definitions start with F₀=0, F₁=1.