— Burt Kaliski Jr. (@modulomathy) February 25, 2012
Replacing 7 with x and 8 with x+1 gives the general formula
x(x+1) = ((x+(x+1))2-1)/4.
Replacing x+(x+1) with 2x+1 on the right hand side gives
x(x+1) = ((2x+1)2-1)/4
which expands then simplifies to
x(x+1) = ((4x2+4x+1)-1)/4
Both sides are consistent, confirming that the pattern is correct.
The pattern can be expressed another way that may be more useful in practice.
So far, we’ve just replaced 7 with x and 8 with x+1. We can instead replace the 8 with y and the -1 with -(x–y)2. We then have
xy = ((x+y)2-(x–y)2)/4.
Expanding the right-hand side then simplifying gives:
xy = ((x2+2xy+y2)-(x2-2xy+y2))/4
= 4xy/4 ,
again confirming the pattern.
This pattern is more useful because it tells us that the product of any two numbers x and y equals one quarter the difference between the square of their sums, and the square of their difference. The example given for 56 is a special case. The product of any two consecutive numbers is one quarter of the difference between the square of their sums, and 1.