Fibonacci Sequence as a Recursive Sequence

A sequence in arithmetic is an association of numbers in a sequential method that follows a particular rule; every time period of a sequence is expounded to its earlier and successive time period by that given rule.

Extra formally, a sequence could also be outlined as a mapping or a operate f: 𝐍→X outlined as                                    f(n) = x_n  for each n in 𝐍

such that each one the x_n; n = 1, 2, 3,… is a sequence in X ruled by the rule f.

Some examples of a sequence are:

• Sequence 2, 6, 12, 30,… the final rule for this specific sequence is n(n + 1); n = 1, 2, 3,…
• 1, 1/2, 1/3, 1/4,… whose normal rule is 1/n the place n is a Pure Quantity.

Recursive Sequence

A recursive sequence is a sequence shaped by a recursive operate. Recursion is a course of through which a sequence is shaped by choosing an preliminary time period to start the sequence and repeatedly utilizing the earlier time period to seek out the following time period. Thus, recursion is a recursive operate that makes use of the preliminary or previous values to get the successive phrases. There are two steps of a recursion:

• Primary Step: Specifies a group of beginning values or preliminary values of the operate.
• Recursive Step: Offers the rule to type new parts primarily based on identified values or earlier values of the sequence.

Typically an exclusion set can also be outlined, which specifies a set of values that aren’t to be included within the recursion course of.

Instance of a recursive sequence:

The sequence of pure numbers.

Primary Step: Let f be the recursive operate whose preliminary worth f_o = 0

Recursive Step: f_n = f_n + 1; In response to recursive rule

                 f₁ = f_o + 1 = 0 + 1

                 f₂ = f₁ + 1 = 2 and so forth we get the sequence of pure numbers

Fibonacci Sequence

A outstanding instance of a recursive sequence is a Fibonacci Sequence. This sequence is confirmed to be one of the vital intriguing and ubiquitous of all quantity sequences in arithmetic. Fibonacci Sequence, also referred to as Fibonacci Numbers, is outlined recursively as:

For n be any quantity, n ≥ 0, if F_n  is the nth Fibonacci quantity, then we’ve got

• Primary Step: The preliminary values of the recursion, F_o= 0 and F₁= 1
• Recursive Step: The recurrence relation is outlined as

                      F_n = F_{n – 1} + F_{n – 2} , n ≥ 2

The successive Fibonacci Numbers are discovered by including the previous two numbers of the sequence. Therefore, the phrases within the sequence are 0, 1, 0 + 1 = 1, 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8, 5 + 8 = 13, that’s, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 and so forth.

Some Properties of Fibonacci Numbers

The quantity sample has some attention-grabbing properties, which makes it such a major quantity sequence.

• Successive numbers of the sequence have a recurrence relation.
• The Biggest Frequent Divisor of successive phrases of the sequence is 1.

For n ≥ 0 and F_n be recursion operate for the sequence, gcd( F_n, F_n+1) = 1. Allow us to examine this property b taking an instance, gcd(F₅, F₆) = gcd(5, 8) = 1 and gcd(F₉, F₁₀) = gcd(34, 55), now components of 34 are 1, 2, 17 and 34 and components of 55 are 1, 5, 11 and 55. Thus, gcd(34, 55) = 1.

• For n ≥ 0, gcd(F_n, F_{n + 2}) = 1

We are able to additionally examine this one by taking an instance, F₅ = 5 and F_{5 + 2} = F₇ = 13, clearly

gcd(5, 13) = 1.

• The sum of any six consecutive Fibonacci Numbers is a a number of of 4.

Allow us to take, F₂ + F₃ + F₄ + F₅ + F₆ + F₇ = 1 + 2 + 3 + 5 + 8 + 13 = 32 = 4 × 8.

• The ratio of any two consecutive Fibonacci Numbers is roughly equal to the Golden Ratio.

The worth of the Golden ratio, φ = 1.618, roughly

F₆ : F₅ = 8 : 5 = 1.6 which is near the Golden Ratio.