1. I have discovered a geometric progression, shown below.
1
0
( )( 1) 1
n
j n
j
x x x
Here is a proof by induction for n:
Basis: Let n=2
Inductive Step: Assume
and show:
by inserting the inductive hypothesis,
then, by the Product of Powers Property, we can add exponents, giving us:
therefore, by the principle of mathematical induction,
for all n>=2.