Question

For 0 < a < 4, define the sequence \(\left\{x_n\right\}^{\infin}_{n=1}\) of real numbers as follows :
x₁ = a and x_n+1 + 2 = −x_n(x_n - 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

The Correct Option is B, C

- (A) is false because the behavior of the sequence does not ensure convergence for at least three distinct values of \( \alpha \in (0, 1) \). - (B) is true. For \( \alpha \in (1, 2) \), the sequence has a tendency to converge for three distinct values as the sequence undergoes sufficient interaction to stabilize. - (C) is true. For \( \alpha \in (2, 3) \), there are more instances where the sequence converges to three distinct values because of the interaction of the recurrence relation. - (D) is false because \( \{x_n\} \) does not converge for three distinct values in \( \alpha \in (3, 4) \). Thus, the correct answers are (B) and (C).