Challenge Level

Janine noticed, while studying some cube numbers, that ``if you
take three consecutive whole numbers and multiply them together and
then add the middle number of the three, you get the middle number
cubed''; *e.g.*, 3, 4, 5 gives 3 x 4 x 5 + 4 = 64, which is
a perfect cube. Does this always work? Can you prove or disprove
this conjecture?