# All Square Numbers Follow a Recursive Series?

• IntegrateRSC
In summary, the conversation discusses a recursive series that can generate every square number. The example given shows how the series works and there is a suggestion to use the Z transform to prove it. It is also mentioned that these sequences correspond to the natural responses of linear discrete-time systems.
IntegrateRSC
Sorry if this is a well known thing, but I've noticed this and decided to see how well known it is, also if there is a way to prove it other than the recursive series.

an=(an-1-an-2+2)+an-1

This recursive series will in fact generate every square number. Take in example:

a0=0
a1=1

So if you use the recursive series above:
a2=(a1-a0+2)+a1
a2=(1-0+2)+1

Any comments? I'm praying I used the 'sub' tags right

let's call sqrt(an-1) is x. Then we want to prove that 2x2=(x+1)2+(x-1)2-2. Simple algebra.

Look up the Z transform and it's inverse. You can form recursive solutions corresponding to polynomials of arbitrary order. These sequences correspond to the natural responses of linear discrete-time systems.

## 1. What is a recursive series?

A recursive series is a sequence of numbers where each number is calculated from the previous number using a specific rule or formula.

## 2. How do you know if a series is recursive?

A series is recursive if each number in the series follows a specific pattern or rule, such as adding a constant number or multiplying by a fixed value.

## 3. Are all square numbers recursive?

Yes, all square numbers follow a recursive series. Each square number is calculated by multiplying the previous number by 2 and adding 1.

## 4. Is there a formula for finding the nth term in a recursive series?

Yes, there is a formula for finding the nth term in a recursive series. It is often called the recursive formula and it involves using the previous term(s) to calculate the next term in the series.

## 5. Can recursive series be used in real-life applications?

Yes, recursive series can be used in real-life applications, such as in computer programming or financial calculations. They are also used in various fields of science, such as physics, biology, and chemistry, to model natural processes.

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