Exploring Reasoning and Proof
 Introduction | Triangular and Square Numbers | Conjecturing with Algebraic Symbols | Proving the Conjecture | Reflection Questions | Summary | Your Journal

In the previous diagram, we showed that 6 + 10 = 16 is a supporting case for the conjecture. How can we work with this in more general terms?

 1. Using numbers: Show Answer
 Sample Answer: 6 + 10 = 16, 10 + 15 = 25, 15 + 21 = 36 etc.
 2. Using words: Show Answer
 Sample Answer: A triangle number + the next triangle number = a square number
 3. Using symbols: Show Answer
 Sample Answer: Substituting the equation for the triangular numbers: n(n+1)/2 + (n+1)(n+2)/2 = a square number.

This is one way to state the conjecture we have to prove. We can further refine it by noting that in our numeric example, 6 + 10 = 15, this relates to the sum (1 + 2 + 3) + (1+ 2 + 3 + 4) = 16, with that insight we can state our conjecture like this, n (n + 1) / 2 + (n + 1) (n + 2) / 2 = (n + 1)2.

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