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
Prove the following:
Let f(x) be a polynomial in R[x] of positive degree n.
1. The polynomial f(x) factors in R[x] as the product of
polynomials of degree
1 or 2.
2. The polynomial f(x) has n roots in C (counting multiplicity).
In particular,
there are non-negative integers r and s satisfying r+2s = n such
that
f(x) has r real roots and s pairs of non-real conjugate complex
numbers as
roots.
3. The polynomial f(x) factors in C[x] as...
Let function F(n, m) outputs n if m = 0 and F(n, m − 1) + 1
otherwise.
1. Evaluate F(10, 6).
2. Write a recursion of the running time and solve it
. 3. What does F(n, m) compute? Express it in terms of n and
m.