#math trivia for #February25: #56 is a product of consecutive numbers: 56 = 7*8. And 56 = ((7+8)^2)-1)/4. What’s the general formula?

— 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) = ((2*x*+1)^{2}-1)/4

which expands then simplifies to

*x*(*x*+1) = ((4*x*^{2}*+4x*+1)-1)/4

= (4x^{2}+4x)/4

= x^{2}+x

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

*x*y = ((*x*+y)^{2}-(*x*–*y*)^{2})/4.

Expanding the right-hand side then simplifying gives:

*xy* = ((*x*^{2}*+*2*x*y+*y*^{2})-(*x*^{2}-2*xy*+*y ^{2}*))/4

= 4

*xy*/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.