Can the Strobe Effect be Explained Through Gauss's Induction Proofs?

[Ben2]

From Gauss, Disquisitiones Arithmeticae p. 10: Given A(1) = a(1) = [a(1)], A(2) = a(2)A(1) + 1 = [a(1), a(2)]and for n>= 3, A(n) = a(n)A(n-1) + A(n-2) = [a(1), a(2), ... , a(n)], prove that [a(1), a(2), ... , a(n)]*[a(2), a(3), ... , a(n-1)] - [a(1), a(2), ... , a(n-1)]*[a(2), a(3), ... , a(n)] = (-1)^n, where "^" indicates "to the power", and that
[a(1), a(2), ... , a(n)] = [a(n), a(n-1), ... , a(1)].
This stems from an interest in congruences, coming in turn from Nishiyama's online article on the strobe effect--a physics question. Thanks to all who comment.

 

[Ben2]

Have tried induction without success. Will appreciate any help.

 

[PeroK]

Have tried induction without success. Will appreciate any help.

Someone else may be familiar enough with the material to help you, but without further explanation I don't know what that means. It would also be better in Latex.

https://www.physicsforums.com/help/latexhelp/

 

[berkeman]

From Gauss, Disquisitiones Arithmeticae p. 10: Given A(1) = a(1) = [a(1)], A(2) = a(2)A(1) + 1 = [a(1), a(2)]and for n>= 3, A(n) = a(n)A(n-1) + A(n-2) = [a(1), a(2), ... , a(n)], prove that [a(1), a(2), ... , a(n)]*[a(2), a(3), ... , a(n-1)] - [a(1), a(2), ... , a(n-1)]*[a(2), a(3), ... , a(n)] = (-1)^n, where "^" indicates "to the power", and that
[a(1), a(2), ... , a(n)] = [a(n), a(n-1), ... , a(1)].
This stems from an interest in congruences, coming in turn from Nishiyama's online article on the strobe effect--a physics question. Thanks to all who comment.

Welcome to the PF, Ben.

Please re-post your question in the technical Math forums, and provide links to the material you want to discuss. Also as suggested, it helps if you post using LaTeX, to make the formatting easier to read. Thanks!