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Recursion is the process of repeating it-self in its own way. The term is very vast and has many implications. If we are taking about practical implementation of recursion we can take two parallel mirrors facing each other, now we will see so many mirrors in those two parallel mirrors thats because of recursion. The process of recursion is basically used in mathematics and computer science.
Recursion in discrete mathematics can be defined as the process of defining a term in terms of itself( or the part of itself) . As Q(n) could be recursive in Q(n+1). there are also recursive definitions, functions, sequences, sets and algos. Recursion in discrete mathematics is a bit strange to understand but once it is in process it can be the most powerful way of expressing some processes.
If we express z as z+1 and z+2 and so on then it is recursion discrete mathematics expressed in its own way of repeating the term but with a little change. Here we an also say recursion discrete mathematics also follows the two specific rules or repetition.
First the term should repeat itself at a particular interval of time .
Second is that the repeating term must have some changes at some specific values if necessary. If change not applicable we do not put it into action.
The recursion that repeats itself at a particular time of interval without any dis-formation is known as linear recursion and the recursion that has some dis-formation or changes at some particular time of interval is called non- linear recursion.
Here recursion is a process of repeating terms in its own way and the nature of recursion of any term is known as recursive or recursive in nature. In mathematics we can create recursive Functions. And we often call these as recurrence Relations or recurrence Set.

Recursively Defined Sequence

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Recursively Defined Sequence can be derived from recursive formula. In recursively defined sequences first term of the sequence is always given, and by using previous value we can find n th term of sequence. We can find terms of recursively defined sequence using first term and we can find n th term of this sequence by adding three in (n - 1)th term. Suppose a sequence where first term a1 = 4 and we have to find a2 and a3, we can find a2 by adding 3 in a1, a2 = a1 + 3 => 4 + 3 = 7. Similarly we can find a3 by adding 3 in a2, a3 = a2 + 3 = 7 + 3 = 10, our sequence is a1, a2, a3 is 4, 7, 10.

Using previous values we can generate n th term but if we want to find the value of 100 th term, to do this we need to find previous term of 100 th term means 99th term, and to find 99 th term we have to find out 98 th term and so on, but this process becomes very lengthy, so we have to derive a formula to find the nth term. Let’s take a series 7, 9, 11, 13,........n, here first term is 7 and we have to derive formula for nth term. So first of all find the difference between first term and second term which is 2 and difference between second and third term is 2, so multiply n th term by 2 = n * 2. To check this formula we will multiply first term by 2, 7*2 = 14, but this is not equals to 9, if we subtract 5 from 14 then we’ll get 9, so we’ll subtract 5 from n th term, our formula becomes, n*2 – 5, using this formula we can find 80th term which is 80 * 2 - 5 => 160 - 5 => 155.