Question

Consider the following expression: 7^n-6*n-1

Using induction, prove the expression is divisible by 36.

I understand the process of mathematical induction, however I do not understand how the solution showed the result for P_n+1 is divisible by 36? How can we be sure something is divisible by 36? Please explain in great detail.

Answer #1

prove by induction that n(n+1)(n+2) is divisible by 6 for
n=1,2...

Prove that 7^(n) − 1 is divisible by 6, for every positvie
integer n

Using induction, prove the following:
i.) If a > -1 and n is a natural number, then (1 + a)^n >=
1 + na
ii.) If a and b are natural numbers, then a + b and ab are also
natural

1. Prove that an integer a is divisible by 5 if and only if a2
is divisible by 5.
2. Deduce that 98765432 is not a perfect square. Hint: You can use
any theorem/proposition or whatever was proved in class.
3. Prove that for all integers n,a,b and c, if n | (a−b) and n |
(b−c) then n | (a−c).
4. Prove that for any two consecutive integers, n and n + 1 we
have that gcd(n,n + 1)...

Without using induction, prove that for x is an odd, positive
integer, 3x ≡−1 (mod 4). I'm not sure how to approach the problem.
I thought to assume that x=2a+1 and then show that 3^x +1 is
divisible by 4 and thus congruent to 3x=-1(mod4) but I'm stuck.

Prove the following statement by mathematical induction. For
every integer n ≥ 0, 2n <(n + 2)!
Proof (by mathematical induction): Let P(n) be the inequality 2n
< (n + 2)!.
We will show that P(n) is true for every integer n ≥ 0. Show
that P(0) is true: Before simplifying, the left-hand side of P(0)
is _______ and the right-hand side is ______ . The fact that the
statement is true can be deduced from that fact that 20...

Prove by mathematical induction that for all odd n ∈ N we have
8|(n2 − 1). To receive credit for this problem, you must show all
of your work with correct notation and language, write complete
sentences, explain your reasoning, and do not leave out any
details.
Further hints: write n=2s+1 and write your problem statement in
terms of P(s).

1) Find the sum S of the series where S = Σ i ai -- here i
varies from 1 to n.
Use the mathematical induction to prove the following:
2) 13 + 33 + 53 + …. + (2n-1)3 = n2(2n2-1)
3) Show that n! > 2n for all n > 3.
4) Show that 9(9n -1) – 8n is divisible by 64.
Show all the steps and calculations for each of the above
and explain your answer in...

Problem 3. Let n ∈ N. Prove, using induction, that Σi^2= Σ(n + 1
− i)(2i − 1). Note: Start by expanding the righthand side, then
look at the following pyramid (see link) from

Prove the following using induction:
(a) For all natural numbers n>2, 2n>2n+1
(b) For all positive integersn,
1^3+3^3+5^3+···+(2^n−1)^3=n^2(2n^2−1)
(c) For all positive natural numbers n,5/4·8^n+3^(3n−1) is
divisible by 19

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