Lesson Make any sequence valid

Source code of 'Make any sequence valid'

This Lesson (Make any sequence valid) was created by by terryriegel(1)

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About terryriegel: I love programming and math.

Whats the next number in a sequence. It can be any number you want.

The solution arises from the fact that anything times zero is just zero and also adding zero to something doesn't change it.

Now granted the formula can get a bit lengthy but it is a valid solution and it can be any number you want.

Lets start with a simple sequence of three random numbers between (1-10) from random.org I got 6,9,2,_ whats next, Ok back to random.org its came up with 5.

So our problem is to come up with a formula f(n) that produces f(1)=6, f(2)=9, f(3)=2, and f(4)=5

f(1)=6
f(2)=9
f(3)=2
f(4)=5

Lets initially choose

{{{f(n) = 6}}}

f(1) = 6 check
f(2) = 6 fail

so lets refine it

{{{f(n) = 6 + (n-1)3/1}}}

f(1) = 6 check
f(2) = 9 check
f(3) = 12 fail

so we refine further

{{{f(n) = 6 + (n-1)3/1 + (n-1)(n-2)(-10)/2}}}

f(1) = 6 check
f(2) = 9 check
f(3) = 2 check
f(4) = -15 fail

so we refine further

{{{f(n) = 6 + (n-1)3/1 + (n-1)(n-2)(-10)/2 + (n-1)(n-2)(n-3)20/(2*3)}}}

f(1) = 6 check
f(2) = 9 check
f(3) = 2 check
f(4) = 5 check

{{{ f(n) = 6/0!+(n-1)*3/1!+(n-1)*(n-2)*(-10)/2!+(n-1)*(n-2)*(n-3)*20/3! }}}

There we have it a generalized method for making any sequence valid.

So the next time your teacher asks you whats next in this sequence

1,2,3,4,?

Tell them it can be anything.