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Ken Ward's Mathematics Pages

Negative Factorials First Difference

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Preamble

Previously, we claimed that
Δk(n)=n·k(n-1) [1.01]

And we proved it for integer n>0

Now we wish to show that 1.01, above, works for negative and zero n too.

Proof

We wish to prove 1.01. We have proved it for integer n >0.
If n=-1, then
Δk(0)=k(0)-k(0)
=0

If n=-1, then
Δk(-1)=(k+1)(-1)-k(-1)
Δk(-1)=(k+1)(-1)-k(-1)  
Δk(-1)=1/(k+2)-1/(k+1)
= -1/ (k+2)(k+1)
=(-1)k(-2)

So 1.01 works for n=-1

We wish to prove that
1 [1.01 repeated]
is true for negative integers.
Let n=-p, where p is a positive integer, so
2 [1.02]

So, the first difference is by definition:
3 [1.03]
4 [1.04]
Expanding the factorials, we have:
5 [1.05]
Extracting a factor:

6 [1.06]
Evaluating gives:

7 [1.07]
Rewriting 1.07 as a factorial:
8 [1.08]
substituting n=-p, we have:
9 [1.09]
Hence, 1.01 is true for negative integers too.




Ken Ward's Mathematics Pages


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