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 = 29^{2} = [(4x7)+1]^{2}?
Solution Commentary: The conjecture for the general case is:
1 + (n1)(n)(n+1)(n+2) = X^{2} = [(n1)(n+2)+1]^{2}.
Can you prove it using algebra?...or using...?
