Question

Let \(\{a_n\}_{n=1}^\infty\) be a sequence of positive real numbers satisfying \[ \frac{8}{a_{n+1}} = \frac{7}{a_n} + \frac{a_n^2}{343}, \quad n \geq 1, \] with \(a_1 = 3\) and \(a_n<7\) for all \(n \geq 2\).
Consider the following statements: \begin{itemize} \item [(I)] \(\{a_n\}\) is monotonically increasing. \item [(II)] \(\{a_n\}\) converges to a value in the interval \([3, 7]\). \end{itemize} Then which of the above statements is/are true?

• A. (I) only
• B. (II) only
• C. Both (I) and (II)
• D. Neither (I) nor (II)

Hint:

When working with recurrence relations, analyzing the behavior of the sequence and solving for the limit can help determine its convergence and monotonicity.

Answer 1

The Correct Option is C

We are given the recurrence relation for the sequence \(\{a_n\}\) as: \[ \frac{8}{a_{n+1}} = \frac{7}{a_n} + \frac{a_n^2}{343} \] Rearranging the equation: \[ a_{n+1} = \frac{8}{\frac{7}{a_n} + \frac{a_n^2}{343}} \] We will now analyze the two statements.
Step 1: Statement (I) - Is \(\{a_n\}\) monotonically increasing?
To check if the sequence is monotonically increasing, we need to determine whether \(a_{n+1}>a_n\). Substituting the recurrence relation: - As \(a_n\) increases, \(\frac{7}{a_n}\) decreases, and \(\frac{a_n^2}{343}\) increases. - This suggests that the denominator of the right-hand side of the equation is increasing, which in turn suggests that \(a_{n+1}\) is increasing as \(n\) increases. Therefore, the sequence \(\{a_n\}\) is monotonically increasing.
Step 2: Statement (II) - Does \(\{a_n\}\) converge to a value in the interval \([3, 7]\)?
We are also given that \(a_n<7\) for all \(n \geq 2\), and we need to check if \(\{a_n\}\) converges to a value within the interval \([3, 7]\). Taking the limit as \(n \to \infty\), let \(L\) be the limit of the sequence: \[ L = \frac{8}{\frac{7}{L} + \frac{L^2}{343}} \] Solving this equation yields a value for \(L\) that lies within the interval \([3, 7]\), confirming that the sequence converges to a value within this range. Thus, both statements (I) and (II) are true.