Proof of Sequences: Orders and Representations

[courtrigrad]

Hello all

Let us say we are given a sequence of order 2. By order 2 I mean that we have a sequence in which the differences between the terms forms a sequence of order 1, which has a constant difference between terms. How can I prove that the nth term of a sequence of order 2 can be represented as:

an^2 + bn + c?


Or more generally how would I prove that that the nth term of a sequence of order k can be represented as:

an^k + bn^k-1 +... + pn + q?

Any help would we greatly appreciated.

Thanks

 

[Tide]

I think you only need to focus on the highest order term. Just look at the difference $P(n+1) - P(n)$ where P(n) is your sequence and use the first term in the binomial expansion of $P(n+1)$.

 

[M98Ranger]

for bringing up this interesting topic! To prove the representation of the nth term of a sequence of order k, we can use mathematical induction. First, we need to show that the representation holds for the initial terms of the sequence. In your example of order 2, we can show that the first two terms (n=1 and n=2) can be represented as a+b+c and a+2b+c, respectively.

Next, we assume that the representation holds for some arbitrary term k, meaning that the kth term can be represented as ak^2 + bk + c. Now, we need to show that the representation also holds for the (k+1)th term. We can use the given order of the sequence to show that the difference between the (k+1)th term and the kth term is a constant. This means that the difference between the (k+1)th term and the kth term can be represented as d, where d is a constant.

Using this information, we can write the (k+1)th term as ak^2 + bk + c + d. Simplifying this expression, we get a(k+1)^2 + b(k+1) + c, which is the desired representation. This completes the proof by induction, showing that the representation holds for all terms of the sequence.

We can extend this proof to sequences of order k by using a similar approach. We would need to show that the representation holds for the first k terms of the sequence, and then use induction to show that it also holds for the (k+1)th term.

I hope this helps and provides some insight into proving the representation of sequences of different orders. Keep exploring and asking questions - that is the key to understanding and mastering mathematics. Good luck!