SOLUTION: will this pattern continue 39 x 41 = 40 x 40 - 1, 59 x 61 = 60 x 60 - 1

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Question 188687: will this pattern continue 39 x 41 = 40 x 40 - 1, 59 x 61 = 60 x 60 - 1
Answer by Edwin McCravy(20066) About Me  (Show Source):
You can put this solution on YOUR website!


Write the pattern as A x B = C x C - 1

The pattern does appears to continue

 (A x  B)  = (C x  C) - 1
--------------------------
 19 x  21 =  20 x  20 - 1     =   399
 39 x  41 =  40 x  40 - 1     =  1599
 59 x  61 =  60 x  60 - 1     =  3599
 79 x  81 =  80 x  80 - 1     =  6399
 99 x 101 = 100 x 100 - 1     =  9999
119 x 121 = 120 x 120 - 1     = 14399

but we must find out for certain if the pattern
continues forever:

The numbers in both C columns are multiples of 20,
so the general expression for them is C = 20n

The numbers in column A are 1 less than the numbers 
in column C, so the general expression for them is
A = 20n-1

The numbers in column B are 1 more than the numbers 
in column C, so the general expression for them is
B = 20n+1

Now we want to find out if it is always true that'


        A x B  = C x C - 1

So we substitute 

(20n-1)(20n+1) = (20n)(20n)-1 

Then we simplify each side:

       400n2-1 = 400n2-1 

so since both sides are the same, regardless
of n, we now know for certain that the pattern 
will continue forever.

Edwin