.
In arithmetic progression, the general formula for the next term is
= + d.
This equation is a recursive equation example.
In arithmetic progression, the formula for the n-th term is
= + d*(n-1).
It is an explicit formula example.
Roughly speaking, explicit formula allows calculating the term or the function explicitly.
Recursive formula allows calculating the next term of a sequence via the previous term (or several previous terms).
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If you want, you may look at this youtube lesson
https://www.youtube.com/watch?v=nH_z9mP--c0
about the same subject. They relate it to Algebra-I.
As more complicated/advanced popular example, there is a Fibonacci sequence.
You may read about it in this Wikipedia article
https://en.wikipedia.org/wiki/Fibonacci_sequence
This sequence was originated in XII century by Italian mathematician Fibonacci
who referred to studying a population of rabbits.
(the true story and applications are much more rich and interesting).
For this sequence, also there are recursive and explicit formulas.
So, arithmetic sequences, geometric sequences and the Fibonacci sequence are the closest examples,
but Fibonacci is just the level of 9th - 10th grades.
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O-o-o! There is one more example, quite simple.
It is the formula for the sum of interior angles of n-sided polygon
= 180*(n-2) degrees.
It is an explicit formula.
The recursive analogue is = + 180.
The recursive formula can be deduced geometrically
by drawing a diagonal in an n-sided polygon, cutting a triangle.