Odd Perfect Squares

In my Discrete Math class students had to find an equation representing the odd perfect squares: 1, 9, 25, 49, 81, …. Some found the explicit equation,

\( {{a}_{n}}={{(2n-1)}^{2}}\). One student found a recursive equation, \({{a}_{n}}={{a}_{n-1}}+8(n-1),\quad {{a}_{1}}=1\). After class I thought that the recursive equation ought to correspond to a visual “proof without words,” and it does:

Here are a few words to go with the proof: We start with the blue square in the middle, that’s the initial condition that the first term is 1. Then we add the 8 purple squares to form the next term in the sequence, 1+8=9, a 3 x 3 square. We then add the 8 yellow rectangles, each composed of 2 squares to form 9+8 x 2=25, a 5 x 5 square bounded by the yellow rectangles. Next come the blue rectangles, to form 25 + 8 x 3 = 49, etc.

This visual proof is reminiscent of summing odd numbers to obtain perfect squares, e.g. :

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