Question

For 0 < a < 4, define the sequence \left\\{x_n\right\\}^{\infin}_{n=1} of real numbers as follows :
x1 = a and xn+1 + 2 = −xn(xn - 4) for n ∈ $\N$.
Which of the following is/are TRUE ?

• A. \left\\{x_n\right\\}^{\infin}_{n=1} converges for at least three distinct values of $a \in (0,1)$
• B. \left\\{x_n\right\\}^{\infin}_{n=1} converges for at least three distinct values of $a \in (1,2)$
• C. \left\\{x_n\right\\}^{\infin}_{n=1} converges for at least three distinct values of $a \in (2,3)$
• D. \left\\{x_n\right\\}^{\infin}_{n=1} converges for at least three distinct values of $a \in (3,4)$

Answer 1

Answer: (B), (C)

\left\\{x_n\right\\}^{\infin}_{n=1} converges for at least three distinct values of $a \in (1,2)$

Answer 2

The correct option is (B) : \left\\{x_n\right\\}^{\infin}_{n=1} converges for at least three distinct values of $a \in (1,2)$ and (C) : \left\\{x_n\right\\}^{\infin}_{n=1} converges for at least three distinct values of $a \in (2,3)$.