February 1st, 2016, 08:14 PM  #31  
Senior Member Joined: Jan 2014 From: The backwoods of Northern Ontario Posts: 371 Thanks: 68  Quote:
1 × 2 = 2 (or in the order that you had it: 2 × 1 = 2) Last edited by Timios; February 1st, 2016 at 08:18 PM.  
October 28th, 2016, 10:51 PM  #32 
Newbie Joined: Jan 2016 From: at home Posts: 2 Thanks: 1 
The '=' is a relation operator which is used for assignment, and although we most commonly assign equality, it doesn't change what it truly is. That said ... It turns out the ‘?’ can be whatever we want it to be! Allow me to illustrate. Take for example, f(x)=(1/40)x^4(13/20)x^3+(291/40)x^2(553/20)x+42, and evaluate f(8), f(7), f(6), f(5), f(3). It turns out that: f(8)=56, f(7)=42, f(6)=30, f(5)=20, f(3)=9. Wait, what sorcery is this? Turns out that although the popular rule f(x)=x(x1), which gives f(x)=6, satisfies the known values in the sequence, that f(x)=(1/40)x^4(13/20)x^3+(291/40)x^2(553/20)x+42 also satisfies them–except with a different value of f(3)! Here’s another one that also works but gives f(3)=12: f(x)=(1/20)x^4(13/10)x^3+(271/20)x^2(543/10)x+84 And here is one where f(3)=π f(x)=(1/120)(π6)x^4(13/60)(π6)x^3+(1/120)(251π1386)x^2+(1/60)(3138533π)x+14(π6) Finally, in general if you want the ‘?’=k, i.e., f(3)=k where k is the value of your choice, then (1/120)(k6)x^4(13/60)(k6)x^3+(1/120)(251k1386)x^2+(1/60)(3138533k)x+14(k6) More details here: https://www.scribd.com/doc/260182194...tarySequences For a nonpolynomial rule see here: Last edited by skipjack; December 13th, 2016 at 09:22 PM. 

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