It doesn't seem true, and here's a counterexample: observe that the first three terms form a perfect square. You can rewrite the equation as
[tex](n+p)^2+q^2=r^2[/tex]
This is basically the Pythagorean theorem applied to a triangle with sides n+p, q and r. For example, pick:
[tex]n=1, p=2, q=4, r^5[/tex]
The expression becomes
[tex]1^2+2\cdot 2\cdot 1+2^2+4^2=5^2 \iff 1+4+4+16=25 \iff 25=25[/tex]
Which is true, even if
[tex]r^2\neq q^2[/tex]
