#math trivia #56 solution

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 -(xy)2.  We then have

xy = ((x+y)2-(xy)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.