If $(a, b, c)$ is a Pythagorean triple, then $(a+1, b+1, c+1)$ is not a Pythagorean triple. Pythagoras stated that there exist positive natural numbers, $a$, $b$ and $c$ such that $a^2+b^2=c^2$. These three numbers, $a$, $b$ and $c$ are collectively known as a
Pythagorean triple. For example, $(8, 15, 17)$ is one of these triples as $8^2 + 15^2 = 64 + 225= 289 = 17^2$. Other examples of this triple are $(3, 4, 5)$ and $(5, 12, 13)$.

Using Proof by Contradiction, show that: If $(a, b, c)$ is a Pythagorean triple, then $(a+1, b+1, c+1)$ is not a Pythagorean triple.


Since I assume that you're new to some Mathematical topics, I'd use basic language. 

First of all, the question itself teaches you that, if $(a,b,c)$ is a Pythagorean triplet, then it means that $a^2 + b^2 = c^2$. So if $\rm (foo_1, foo_2, foo_3)$ is a Pythagorean triplet, then $\rm (foo_1)^2 + (foo_2)^2 = (foo_3)^2$ (right?). So as you are given in the question, if $(a+1,b+1,c+1)$ is a triplet,then $(a + 1)^2 + (b + 1)^2 = (c + 1)^2$

You are left with two equations:$$\begin{align}  a^2 + b^2 &= c^2 \\ (a + 1)^2 + (b + 1)^2 &= (c + 1)^2\end{align}$$Subtract, and see the magic. Hint: knowledge of "even," and "odd" numbers will help. Especially that odd numbers are in the form $2m + 1 $ and even ones in $2m$... $m$  is an integer by the way.
