# Ken Ward's Mathematics Pages

Series Contents

## 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
[1.01 repeated]
is true for negative integers.
Let n=-p, where p is a positive integer, so
[1.02]

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

[1.06]
Evaluating gives:

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

Ken Ward's Mathematics Pages

# Faster Arithmetic - by Ken Ward

Ken's book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your understanding of numbers. Paperback and Kindle: