Question

Let the limits of the sequences \(\{x_n\}_{n\ge1}\) and \(\{y_n\}_{n\ge1}\) be \(\lambda\) and \(\lambda^3\) respectively. If the interleaved sequence \(x_1, y_1, x_2, y_2, x_3, y_3, \dots\) has a limit, then its value is ____.

• A. \(\lambda^3\)
• B. \(\lambda+\lambda^3\)
• C. 0 or 1
• D. \(-1\), \(0\) or \(1\)

Hint:

Both x_n and y_n are subsequences of the merged sequence, so they must share its limit; solve lambda = lambda cubed.

Answer 1

The Correct Option is D

Step 1: Use the subsequence principle. If a sequence converges to a limit \(L\), then every subsequence of it also converges to \(L\). The sequence \(x_1,y_1,x_2,y_2,\dots\) has \(\{x_n\}\) and \(\{y_n\}\) as subsequences, the odd-indexed and even-indexed terms.

Step 2: Apply this. If the combined sequence converges to \(L\), then \(x_n \to L\) and \(y_n \to L\) as well. But \(x_n \to \lambda\) and \(y_n \to \lambda^3\) are given. By uniqueness of limits, \(L=\lambda\) and \(L=\lambda^3\), forcing \[\lambda=\lambda^3.\]

Step 3: Solve the equation. \[\lambda^3-\lambda=0 \implies \lambda(\lambda-1)(\lambda+1)=0 \implies \lambda \in \{-1,0,1\}.\]

Step 4: Identify the limit value. Since \(L=\lambda\), the possible values of the limit of the combined sequence are \(-1\), \(0\), or \(1\).

\[\boxed{-1,\ 0,\ \text{or } 1}\]