Help with sequences

[HLN_Rubi]

1) Let U_n+2 = 1/(U_n+1+U_n) with U₁ and U₂ = 1
i) Prove that U_n converges to the square root of 0.5 as n gets larger

Now let U_n+3 = 1/(U_n+2+U_n+1+U_n) with U₁ and U₂ and U₃ = 1
ii) Prove that U_n converges to the square root of 1/3 as n gets larger

iii) Prove or disprove that U_n converges to the square root of (1/k) where k is the U_n+k = 1/(U_n-1 + U_n-2 .. for k times)

 

[Ishuda]

1) Let U_n+2 = 1/(U_n+1+U_n) with U₁ and U₂ = 1
i) Prove that U_n converges to the square root of 0.5 as n gets larger

Now let U_n+3 = 1/(U_n+2+U_n+1+U_n) with U₁ and U₂ and U₃ = 1
ii) Prove that U_n converges to the square root of 1/3 as n gets larger

iii) Prove or disprove that U_n converges to the square root of (1/k) where k is the U_n+k = 1/(U_n-1 + U_n-2 .. for k times)

Prove
(a) U_n is bounded below
(b) U_n is decreasing
Therefore U_n converges.

If U_n converges, U_n+M; M = 0, 1, 2, 3, ... converges and converges to the same thing. Therefore
\(\displaystyle lim_{n \gt \infty} U_{n+3} = U = lim_{n \gt \infty} \frac{1}{U_{n+2} + U_{n+1} + U_m} = \frac{1}{U + U + U} = \frac{1}{3 U}\)
Therefore
U² = \(\displaystyle \frac{1}{3}\)
or
U = \(\displaystyle \frac{1}{\sqrt{3}}\)