Methods of Proof

Prove by induction that $\displaystyle \sum_{r=1}^{n}r(3r - 1) = n^2(n + 1)$ , $\displaystyle \forall n \in \mathbb{N}.$

Prove by induction that

$\displaystyle \sum_{r=1}^{n}r!r = (n+1)! - 1$

for all positive integers $\displaystyle n.$

(a)Consider the statement:
For all integers $\displaystyle a$ and $\displaystyle b$, if $\displaystyle a < b$ then $\displaystyle a^2 < b^2.$
Find a counterexample to show that the statement is false.

(b)Let $\displaystyle n$ be an odd integer.
Prove directly that $\displaystyle n^2 - 1$ is divisible by 4.

Consider the statement: For all odd numbers $\displaystyle n$, $\displaystyle n^2 + 4$ is prime.

(a)Find a counterexample to show that the statement is false.

(b)Prove directly that the difference between the cubes of any two consecutive integers is not divisible by 3.

Prove by induction that, for all positive integers $\displaystyle n$,

$\displaystyle \sum_{r=1}^{n} 2^{r-1}r = 2^n(n-1)+1.$

Consider statements A and B below.
For each statement: if true, provide a proof; if false, provide a counterexample.

A: The sum of the squares of any two consecutive integers is always prime.
B: The sum of the squares of any two consecutive integers is always odd.

Prove by induction that, for all positive integers $\displaystyle n$,

$\displaystyle \sum_{r=1}^{n} \frac{1}{(2r + 1)(2r - 1)} = \frac{n}{2n + 1}.$

Prove by induction that $\displaystyle 7^{n}+2$ is divisible by 3 for all $\displaystyle n\in\mathbb{N}.$

Let $\displaystyle r$ be a positive real number and consider the following statement:

If $\displaystyle r$ is irrational then $\displaystyle \sqrt{r}$ is irrational.

(a)Write down the contrapositive of the statement.

(b)Hence prove that the statement is true.