Prove these scenarios by mathematical induction:
(1) Prove n2 < 2n for all integers
n>4
(2) Prove that a finite set with n elements has 2n
subsets
(3) Prove that every amount of postage of 12 cents or more can
be formed using just 4-cent and 5-cent stamps
a) Prove by induction that if a product of n polynomials is
divisible by an irreducible polynomial p(x) then at least one of
them is divisible by p(x). You can assume without a proof that this
fact is true for two polynomials.
b) Give an example of three polynomials a(x), b(x) and c(x), such
that c(x) divides a(x) ·b(x), but c(x) does not divide neither a(x)
nor b(x).
Use induction to prove that 2 + 4 + 6 + ... + 2n = n2 + n for n
≥ 1.
Prove this theorem as it is given, i.e., don’t first simplify it
algebraically to some other formula that you may recognize before
starting the induction proof.
I'd appreciate if you could label the steps you take, Thank
you!