Fibonacci series is a unique sequence of integers starting with 1.
Each element in the series is sum of the two previous numbers.
The second element is also 1 since there is only one previous element.
The 3rd element is sum of first two elements i.e., 1 + 1 = 2.
The fourth element is the sum of second and third elements i.e., 1 + 2 = 3.
And so on.
Thus the complete series is
1, 1, 2, 3, 5, 8, 13, 21, 34, ........
It is an infinite sequence having no end !
Now what is the relation between Golden Ratio and Fibonacci series ?
Let us find the ratios of two successive numbers in the Fibonacci series.
Ratio of 1st and 2nd nos = 1/1 = 1
Ratio of 3rd and 2nd nos = 2/1 = 2
Ratio of 4th and 3rd nos = 3/2 = 1.5
Similarly 5/3 = 1.66..., 8/5 = 1.6, 13/8 = 1.625, 21/13 = 1.61538......... and so on.
Though the values do not decrease monotonically (continuosly decreasing), it can be seen that "the ratio of the consecutive terms in the Fibonacci series converges to the Golden Ratio" i.e., becomes closer and closer to 1.618...