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#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) = ((2x+1)²-1)/4

which expands then simplifies to

x(x+1) = ((4x²+4x+1)-1)/4
= (4x²+4x)/4
= x²+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

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.