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:
This is Julia's proof:
( x - 1) ( x + 1) = x 2 -
( 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.