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
12 pts) Use Mathematical Induction to prove that
an=n3+5n is divisible by 6 when ever
n≥0. You may explicitly use without proof the fact that the product
n(n+1) of consecutive integers n and n+1 is
always even, that is, you must state where you use this fact in
your proof.Write in complete sentences since this is an induction
proof and not just a calculation. Hint:Look up Pascal’s
triangle.
(a)
Verify the initial case n= 0.
(b)
State the induction hypothesis....
Prove by induction on n that the number of distinct handshakes
between n ≥ 2 people in a room is n*(n − 1)/2 .
Remember to state the inductive hypothesis!
1. Use induction to prove that Summation with n terms where i=1
and Summation 3i 2 − 3i + 1 = n^3 for all n ≥ 1.
2. Let X be the set of all natural numbers x with the property
that x = 4a + 13b for some natural numbers a and b. For example, 30
∈ X since 30 = 4(1) + 13(2), but 5 ∈/ X since there’s no way to add
4’s and 13’s together to...