Julia of Downe House School gave the neatest solution to this problem by substituting 'x-1', 'x', 'x+1' for the three consecutive numbers and giving the following statement of Janine's conjecture:

( x - 1) x ( x + 1) + x = x ³ .

This is Julia's proof:

( x - 1) ( x + 1) = x ² - 1
and
( x ² - 1) x = x ³ - x .

Therefore ( x - 1) x ( x + 1) + x = x ³ .

So Janine's conjecture will always work whichever three consecutive numbers are chosen.