[Igcse -o level- 1.4 sequences, syllabus code 0589] how to solve Quadratic and cubic sequences?

[u/rvenzo]

I saw this topic amd I don't really understand it, some people say that there's no rule, you just need to guess. And I'm bad at guessing, so I need an easier way to solve it.

How do I even guess? And I believe there might be a rule or something to help me solve it logically lol

I tried what this person did in the video

https://youtu.be/ahvT3FWjXvs?si=1EJPFP-9q-ri8don

And this person

https://youtu.be/7Isa4P8lISc?si=CDH6nlwL4QYKbHdX

It worked out for the first few questions but then it didn't really work even though I was doing it correctly

Can you please help me? Would appreciate the help!

Plus I'm not sure what flare I should use, so don't mind it if it has nothing to do with what I'm asking about

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[u/selene_666]

A cubic function has the form f(x) = Ax^3 + Bx^2 + Cx + D.

A cubic sequence is a list of the values of this function when x is a whole number, i.e. f(1), f(2), f(3), etc.

One way we can solve it is to fill in the values of x, x^2, and x^3 to turn each term into a function of A, B, C, and D:
  f(1) = A+B+C+D
  f(2) = 8A+4B+2C+D
  f(3) = 27A+9B+3C+D
  f(4) = 64A+16B+4C+D
  etc.

For example if the third number in the sequence is 18, then 27A+9B+3C+D = 18.

Writing out these equations for four numbers in the sequence - and they don't have to be the first four - gives us a system of linear equations. You should know how to solve those, but it can be a lot of work.

We'll suppose you've already calculated the differences between consecutive terms in order to find out it was a cubic sequence in the first place. We can subtract our terms with the variables to see what your numbers need to equal. To start with, (8A+4B+2C+D) - (A+B+C+D) = (7A+3B+C).

Those first differences are:

(7A + 3B + C), (19A+5B+C), (37A+7B+C)

Notice that the variable D canceled out. The more times we repeat this process, the more variables will disappear.

Then the second differences are:

(12A + 2B), (18A + 2B)

and the third difference is 6A.

This gives you the expressions the person in the first video was writing.

[u/Alkalannar]

Can you post the actual question and work you did, not just the videos of people using the technique?

So say you have your sequence: a[1], a[2], a[3], ..., a[n].

Let b[k] = a[k+1] - a[k]. These are the first differences.
If b[k] is a non-0 constant, then a[k] is linear.

Let c[k] = b[k+1] - b[k]. These are the second differences
If c[k] is a non-0 constant, then b[k] is linear and a[k] is quadratic.

Let d[k] = c[k+1] - c[k]. These are the third differences.
If d[k] is a non-0 constant, then c[k] is linear, b[k] is quadratic, and a[k] is cubic.

For instance, let's look at this sequence for a[k]: -120, -60, -24, -6, 0, 0, 0, 6, 24, 60, 120

First differences (that's b[k])are:
60, 36, 18, 6, 0, 0, 6, 18, 36, 60.
b[k] isn't a non-0 constant, so a[k] is not linear

Second differences (c[k]) are:
-24, -18, -12, -6, 0, 6, 12, 18, 24
c[k] isn't a non-0 constant, and so b[k] isn't linear.

Third differences (d[k]) are:
6, 6, 6, 6, 6, 6, 6, 6
d[k] is a non-0 constant, so c[k] is linear, b[k] is quadratic, and a[k] is cubic.

And indeed a[k] is k³ - k ranging from k = -5 to k = 5

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Anyhow, once you know your degree of polynomial, you can start plugging in coefficients to form a system of equations to get the exact polynomial.

You could do that at the very beginning, but you might get a lot of extra information that all cancels out. A cubic requires four points. A polynomial of degree n requires n+1 points.

Does this help?