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Good afternoon. Yesterday at school, I created an equation, which although it is not anything out of the world, but is my first equation for myself.

I thought:

What do I do to create an equation?

A: // Assign an algebraic term to each object that varies.

This equation is for simple succession such as:

    9,5,1,-3,-7  Problem: find the sixth number of the successon.

Pattern: -4 | My equation: -(p * c) + (p + i) 

Where: 
p = pattern of succession
c = The position of number that we want find (e.g. sixth)
i = The first number of succession

So: -(4 * 6) + (4 + 9) -> -18 + 12 = -24 + 13 = -11

And it work perfectly, for all cases( i think), that made me quite happy.

Also, I also did it for the 4 basic operations:

Multiplication case: (p^c) * (i/p) e.g. pattern = n * 4 

6,24,96, ? -> c = 4 , i = 6 , p = 4 ==>  (4^4) * (6/4) = 384


Division case: (p^-c) * (i*p)

Well, I like it. But how could it be made for my equation to be valid for any succession pattern ?, I mean imagine that the pattern is now something more complicated like:

a (n) = n * 2 + !n

Any ideas ?, what I've done, as it was originally called? Is there the actual equation for this? Please some answers for me are difficult to understand, because I have only knowledge of the school, the eleventh grade.

ESCM
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1 Answers1

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what you call a pattern is known as a recurrence relation. They relate successive terms in a sequence of numbers. The usual way to write this is to denote by $x_n$ the $n$-th number in the sequence, then the next number is $x_{n+1}$ and the previous number is $x_{n-1}$.

In your first sequence, the recurrence relation is $$x_{n+1} = x_{n} + 4$$ which means that the $n+1$-th term is obtained from the $n$-th one by adding 4. This is called an arithmetic sequence and one can prove that $$x_n = x_0 + 4 n$$

The second sequence satisfies $$x_{n+1} = 4 x_n$$ This is a geometric sequence, it satisfies $$x_n = 4^n x_0$$

The third recurrence relation is

$$x_{n+1} = 2 x_{n} + x_n !$$

Most recurrence relations cannot be solved by giving a closed form for $x_n$. It is very plausible that this one falls into this category. But really I don't know how to handle it.

Gribouillis
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