# Interesting arithmetic sequence

1. Jun 29, 2011

### xholicwriter

Given N= 1.2.3 + 2.3.4 + ... + n(n+1)(n+2), prove that 4N + 1 is a square (n is a positive integer)

2. Jun 29, 2011

### micromass

It's easy to see that (see www.math.uic.edu/~kauffman/DCalc.pdf on how to find such things)

$$1.2.3+...+n(n-1)(n-2)=\frac{(n+1)n(n-1)(n-2)}{4}$$

Thus it suffices to show that

$$(n+1)n(n-1)(n-2)+1$$

is a square. Which is a much simpler problem.

3. Jun 29, 2011

Great!