Re: How to prove the following for Φ^n

On Mar 28, 10:18 pm, Alpha <vcpan...@xxxxxxxxx> wrote:
How to prove the following:
---------------------------------------
Lim [n→∞] Φ^n = Decimal part converges to 0 i.e. we have +ve INTEGER
Infinity

Nope, the decimal part of Φ^n does not converges to 0,
only the sub-sequence for odd n converges to 0,
the sub-sequence for even n converges to 1.

Here goes my Idea which may or may not be useful:
For an even power : Φ^2k + (1/Φ)^2k = a Whole Number
For an odd power : Φ^(2k+1) - (1/Φ)^(2k+1) = a Whole Number

Hint: To see how to use the two ideas above.
Compute the first few Φ^n and (1/Φ)^n. Can you
observe the pattern between decimal part of Φ^n
and (1/Φ)^n?

.

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