A modified Fibonacci sequence is generated from the...

A modified Fibonacci sequence is generated from the recursion relation ai+1 = ai – ai-1 a) Show that if one writes a vector ui = (ai / ai+1) then the nth term in the sequence is given by un = An u0 where A is the matrix ( 0 1) (-1 1) b) Show that the eigenvalues of the matrix A are given by ? = e±i?/3 c) Diagonalize the matrix and hence show that An = 1/sin ?/3 (-sin(n-1)?/3 sin n?/3) (-sin n?/3 sin(n+1)?/3) d) From your previous results, show that A6 is the identity matrix, and hence explain why this modified Fibonacci sequence will repeat after 6 iterations, no matter what the starting numbers are.

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00:01 Hello everyone, your most welcome to this particular session here in this question given that ai plus 1 equal to ai minus ai minus 1 where a1 equal to 1 and a2 equal to 2.

00:13 Now we have to find a recursion formula for this series.

00:18 So what we have to find? just we find the all term like a1, a2, a3, a4 to get a recursion formula.

00:26 So let us take a3.

00:30 If we put i equal to 2 then we get a 3 which is equal to a 2 minus a 1 so the value of a 2 is 2 and value of a 1 is 1.

00:45 So a 2 minus 1 is 1 now if you put i equal to 3 so what we get we get a 4 for which a 3 minus a 2 so the value of a 3 is 1 what is the value of a 3 1? so 1 minus 2 which is minus 1.

01:04 Now we have to find a 5 so we get a 4 minus a 3 so the value of a 4 is minus 1 and the value of a 3 is 1.

01:16 So 1 minus 1 which is equal to minus 2.

01:19 Now similarly we have to find a 6 so what we get a 5 minus a 4 so value of a 5 is minus a 4 so value of a 5 is minus 2 and minus value of a4 is minus 1.

01:34 So this will become minus 2 plus 1.

01:37 So minus 1.

01:38 Now similarly a7 for which a6 minus a 5.

01:44 So the value of a6 is minus 1 and the value of a5 is minus 2.

01:49 So this will become 1.

01:51 Now a8.

01:54 So a7 minus a6.

01:58 So the value of a7 is 1 and the value of a6 is minus 1.

02:02 So the value of a6 is minus 1.

02:02 So, which is equal to 2.

02:10 Now, we have to find a9.

02:13 So, a9 equal to a8 minus a7.

02:18 So the value of a8 is 2 minus the value of a7 is 1.

02:22 2 minus 1, which is equal to 1.

02:24 Similarly, a10...

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