For every real r ≧ 0, there is a
sequence {bn(r)} defined by
![(r) (r) n∏ (r)
b0 = 1, bn+1 = bi + r for n ≧ 0.
i=1](a110x.png) | (1) |
These sequences were considered previously, in [1], for integer values of r, and it was
shown that there is a constant 𝜃 = 𝜃(r) such that
![(r) 2n
bn+1 ∼ 𝜃 , n → ∞,](a111x.png) | (2) |
for each r = 1,2,3,⋯ . It was observed that
![b(n2+)1 = 22n + 1, n ≧ 0,](a112x.png) | (3) |
whereby 𝜃(2) = 2, and the problem was proposed “to determine the algebraic or
transcendental character of the real numbers 𝜃(r) for r = 1,3,4,5,6,⋯.”
In this paper, we observe explicitly (in §II) that
![n n
b(4n)= τ2 + τ− 2 + 2, n ≧ 1,](a113x.png) | (4) |
where τ = ( + 1)∕2 = 1.618⋯ is the “Golden Mean”, and thus
𝜃(4) = τ2 = ( + 3)∕2 = 2.618⋯ .
|