Fibonacci Calculator

How does this Fibonacci calculator work?

In mathematics, the Fibonacci sequence is defined as a number sequence having the particularity that the first two numbers are 0 and 1, and that each subsequent number is obtained by the sum of the previous two terms. Fibonacci formula:

f₀ = 0     f₁ = 1
f_n = f_n-1 + f_n-2

To figure out the n^th term (x_n) in the sequence this Fibonacci calculator uses the golden ratio number, as explained below:

Φ (phi) = (1+√5)/2 = 1.6180339887

x_n=[1.6180339887ⁿ – (-0.6180339887)ⁿ]/√5

To determine the sum of all numbers until the ^nth term within the Fibonacci sequence first you should calculate the (n+2)^th term in the sequence and then subtract 1 from it:

Sum until the n^{th term} = f_n+2 - 1

Example of a calculation

Assuming we want to figure out the 25^th number in the Fibonacci sequence and then find out the sum of all numbers until 25^th term:

25^th = 75025

Sum until the 25^th term = 196417

This is because the 27^th (25+ 2)^th number is 196418, and if we subtract 1 from it we will get the right value 196417.