Math Patterns: “Can you find the rule for this sequence?”
Question
10, 22, 36, 55, …
Answer
One answer is (n^3 – 4n^2 + 29n – 6) / 2 for n=1 to 4.
How to get it?
10 22 36 55
We have four “numbers”. We can assign a polynomial of what degree(?) in order to form simulteneous equations for which we can ‘solve’?… I mean which polynomial has ‘4’ unknowns since we have ‘4’ answers above that can be used to form 4 simultaneous equations? Its the polynomial of degree 3 which has ‘4’ unknowns! Indeed,
P(n) = an^3 + bn^2 + cn + d where a,b,c,d are constants and need to be determined.
By setting,
P(1) = 10, P(2) =22, P(3) =36, and P(4) = 55, we get 4 equations with 4 unknowns and we can solve it…
I omitted the long tedious details, and we find that,
a = 1/2 , b= -2, c = 29/2 and d = -3.
giving,
(n^3 – 4n^2 + 29n – 6) / 2 = P(n)
That’s it.
No guess work.. It can be reasoned out.. That’s what i attempt to show here. You can tackle many similar integer sequence problems in the same manner. Instead of just giving you a “formula” I believe I wanted to share one way of how to find it.. there are other ways [ for example, the method of finite differences which is actually the way above but cut downs a lot of “tedious calculations” via its theory ].. but it suffices for now.. 🙂
Because, ” Give a man a fish and he will eat for a day, but teach him how to fish, and he will eat for the rest of his days”.
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