Fibonacci Sequence Formula

About Fibonacci Sequence Formula

Setting F0 = 0, F1 = 1, and using the formula below to determine Fn, the Fibonacci sequence formula for "F_n" is defined using the recursive formula. The Fibonacci formula can be found here. F_n = F_n-1 + F_n-2, where n > 1

It's worth noting that F0 is the first phrase here (but NOT F1).

Properties of Fibonacci Sequence The Fibonacci sequence has the following intriguing properties:

1. The golden ratio is connected to Fibonacci numbers. The golden ratio can be used to calculate any Fibonacci number, Fn =(Φn - (1-Φ)n)/√5, φ is golden ratio and Φ ≈ 1.618034. find 7th term, we apply F7 = [(1.618034)7 - (1-1.618034)7] / √5 = 13
2. The "golden ratio" is the ratio of successive Fibonacci numbers. Let A and B be the Fibonacci sequence's two consecutive numbers. B/A thus approaches the Golden Ratio. We may use the formula above to discover any term in the Fibonacci sequence.
3. The approximated Fibonacci number is obtained by multiplying the preceding Fibonacci number by the golden ratio (1.618034). 13 is a number in the sequence, for example, and 13 × 1.618034... = 21.034442. The next Fibonacci number after 13 in the sequence is 21.
4. A multiple of n is every nth number. Examine the series for another pattern of interest. Every third number in the sequence is a two-digit multiple. Every fourth number is a multiple of three, and every fifth number is a multiple of five.
5. The Fibonacci sequence can also be used below zero. F_-n = (-1)ⁿ⁺¹ Fn is the formula. For instance, F_-4 = (-1)⁵. F₄ = (-1)³ = -3.
6. The sum of n terms of Fibonacci Sequence is given by Σi = 0n Fi = Fn+2 - F2 (or) Fn+2 - 1, where Fn is the nth Fibonacci number. (Note: the first term starts from F0)
   For example, the sum of first 10 terms of sequence = 12th term - 1 = 89 - 1 = 88. It can be mathematically written as Σi=09 Fi = F11- 1 = 89 - 1 = 88.

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