Comments on Picture proof for expansion of $x^n−y^n$
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Picture proof for expansion of $x^n−y^n$
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Most students fail to intuit $x^n−y^n \equiv (x−y)(x^{n−1}+x^{n−2}y+...+xy^{n−2}+y^{n−1})$ as substantiated by the glut of duplicates, at least 20 on Math StackExchange. Thus how can students pictorialize it? I seek **solely **VISUAL (not algebraic) proofs here.
After substituting $z = \dfrac xy$, above identity follows from $z^n−1 \equiv (z−1)(z^{n−1}+z^{n−2}+...+z+1)$.
I couldn't find a picture proof from Roger B. Nelson's Proofs without Words (1993),
Proofs without Words II (2000), or
Proofs without Words III (2015).
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