Please help

Might you mean difference equation?

You do realize that this is a help forum where we help students solve their math problems. If you show no work then no one here knows what type of help you need and this is why no one has responded to your post.
 
Mark Behrens said:
3, -6, 12, 4, 20, ?
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This appears to be just a sequence to extend, or a set to complete, not a differential equation or a difference equation. The fact is that, without further information, the next number could be absolutely anything, since any list of numbers is a valid sequence. I can see a slightly pretty pattern, but that wouldn't necessarily be what's expected.

So, is there any context to this question from which we could determine what restrictions to assume?
 
We could set this up as (0, 3), (1, -6), (2,12), (3, 4), and (4, 20). There are, of course, infinitely many functions that satisfy this as there are infinitely many sequences that start "3, -6, 12, 4, 20". But, since there are 5 "data points" there is a unique fourth degree polynomial that will fit.

Write that polynomial as y= ax^4+ bx^3+ cx^2+ dx+ e. Then the five data points give five equations to solve for the five values of a, b, c,, d, and e:
(0, 3): 3= e
(1, -6): -6= a+ b+ c+ d+ e
(2, 12): 12= 16a+ 8b+ 4c+ 2d+ e
(3, 4): 4= 81a+ 27b+ 9c+ 3d+ e
(4, 20): 20= 256a+ 64b+ 16c+ 4d+ e

Solve those equations for a, b, c, d, and e then calculate a(5)^4+ b(5)^3+ c(5)^2+ d(5)+ e for the next number in the sequence.

(Again, that is one out of infinitely many next terms.)
 
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