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Patterns Pop Up Everywhere!

Complete the following:

1 + 0x1x2x3 = ?

1 + 1x2x3x4 = ?

1 + 2x3x4x5 = ?

1 + 3x4x5x6 = ?

1 + 4x5x6x7 = ?

1 + 5x6x7x8 = ?

...........

Formulate a conjecture for the general case based on the observed patterns...be sure to look carefully as there are actually two patterns going on.

Now, try to "prove" your conjecture...

 


Hint: Did you see that:

1 + 4x5x6x7 = 841 = 292 = [(4x7)+1]2?

 


Solution Commentary: The conjecture for the general case is:

1 + (n-1)(n)(n+1)(n+2) = X2 = [(n-1)(n+2)+1]2.

Can you prove it using algebra?...or using...?