To solve the given problem, we have sequences $\{a_n\}$ and $\{b_n\}$ defined recursively as:

• Initial values: $a_1 = 2, b_1 = 1$
• For $n \geq 1$: $a_{n+1} = \frac{a_n + b_n}{2}, \quad b_{n+1} = \frac{2a_n b_n}{a_n + b_n}$

We need to show three things:

1. $b_n \leq a_n$ for all $n$.
2. $b_{n+1} \geq b_n$ for all $n$.
3. Both sequences $\{a_n\}$ and $\{b_n\}$ converge to the same limit, $\sqrt{2}$.

Proof:

(i) Show that $b_n \leq a_n$ for all $n$:

We start by induction.

• Base case: For $n = 1$, we have $a_1 = 2$ and $b_1 = 1$. Clearly, $b_1 \leq a_1$.

• Inductive step: Suppose $b_n \leq a_n$ for some $n$. Now, we need to show that $b_{n+1} \leq a_{n+1}$.

From the recursive definitions:

$a_{n+1} = \frac{a_n + b_n}{2}, \quad b_{n+1} = \frac{2a_n b_n}{a_n + b_n}$

Since $b_n \leq a_n$, $a_n + b_n \geq 2b_n$. This implies:

$\frac{a_n + b_n}{2} \geq b_n$

Also, for $b_{n+1}$:

$b_{n+1} = \frac{2a_n b_n}{a_n + b_n} \leq \frac{2a_n b_n}{2b_n} = a_n \quad \Rightarrow \quad b_{n+1} \leq a_{n+1}$

Thus, by induction, $b_n \leq a_n$ for all $n$.

(ii) Show that $b_{n+1} \geq b_n$ for all $n$:

We need to show that $b_{n+1} - b_n \geq 0$.

$b_{n+1} = \frac{2a_n b_n}{a_n + b_n}$

Consider the difference:

$b_{n+1} - b_n = \frac{2a_n b_n}{a_n + b_n} - b_n = \frac{2a_n b_n - b_n(a_n + b_n)}{a_n + b_n}$ $b_{n+1} - b_n = \frac{2a_n b_n - a_n b_n - b_n^2}{a_n + b_n} = \frac{a_n b_n - b_n^2}{a_n + b_n}$ $b_{n+1} - b_n = \frac{b_n(a_n - b_n)}{a_n + b_n}$

Since $a_n \geq b_n$, it follows that $a_n - b_n \geq 0$, and hence $b_{n+1} \geq b_n$.

(iii) Show that both sequences $\{a_n\}$ and $\{b_n\}$ converge to the same limit $\sqrt{2}$:

Let $L$ be the common limit of $\{a_n\}$ and $\{b_n\}$.

• As $n \to \infty$, $a_n \approx a_{n+1} \approx L$ and $b_n \approx b_{n+1} \approx L$.

From the recursive definitions:

$L = \frac{L + L}{2} = L \quad \text{and} \quad L = \frac{2L^2}{L + L} = \frac{2L^2}{2L} = \sqrt{2}$

Thus, both sequences converge to $\sqrt{2}$.

If you'd like a more detailed breakdown of any part or have questions, let me know!

Relative Questions:

1. What other methods can be used to analyze the convergence of recursive sequences?
2. Can we generalize this approach for other initial values of $a_1$ and $b_1$?
3. How does the arithmetic mean-geometric mean inequality relate to these sequences?
4. What happens if we modify the recurrence relation for $b_{n+1}$?
5. How can this convergence be visualized graphically?

Tip:

When analyzing recursive sequences, always start by establishing bounds and then investigate monotonicity and convergence properties.