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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

f_{1} = f_{o }+ 1 = 0 + 1

f_{2} = f_{1} + 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 F_{1}= 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(F_{5}, F_{6}) = gcd(5, 8) = 1 and gcd(F_{9}, F_{10}) = 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, F_{5} = 5 and F_{5 + 2} = F_{7} = 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, F_{2} + F_{3} + F_{4} + F_{5} + F_{6} + F_{7} = 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

F_{6} : F_{5} = 8 : 5 = 1.6 which is near the Golden Ratio.

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