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Recurrence with a square root
Task number: 3876
For the sequence given by the recurrence a0=2,a1=8,an+2=√anan+1 determine lim.
Solution
We introduce a sequence b_n=\log_2 a_n.
It holds that b_0=1, b_1=3 and b_{n+2}=\frac{b_{n+1}+b_n}2.
According to the previous task, \lim_{n\to\infty} b_n=\frac{b_0+2b_2}3=\frac73.
Answer
The limit is 2^{7/3}=4 {\cdot} 2^{1/3}.