Let $\{a_{n}\}$ and $\{b_{n}\}$ be sequences of real numbers defined as $a_{1}=1$ and for $n\ge1$, $a_{n+1}=a_{n}+(-1)^{n}2^{-n},b_{n}=\frac{2a_{n+1}-a_{n}}{a_{n}}.$ then:

Question

Accepted Answer

To determine why Option 1 is the correct answer, we need to analyze the sequences $\{a_n\}$ and $\{b_n\}$.

### Analysis of $\{a_n\}$:

The sequence $\{a_n\}$ is defined recursively by:
- $a_1 = 1$
- $a_{n+1} = a_n + (-1)^n 2^{-n}$

Let's examine the first few terms to identify a pattern:
- $a_1 = 1$
- $a_2 = a_1 + (-1)^1 \cdot 2^{-1} = 1 - \frac{1}{2} = \frac{1}{2}$
- $a_3 = a_2 + (-1)^2 \cdot 2^{-2} = \frac{1}{2} + \frac{1}{4} = \frac{3}{4}$
- $a_4 = a_3 + (-1)^3 \cdot 2^{-3} = \frac{3}{4} - \frac{1}{8} = \frac{5}{8}$

The sequence $\{a_n\}$ alternates between adding and subtracting increasingly smaller powers of 2. This suggests that $\{a_n\}$ is converging to a limit. To find this limit, observe that the terms are bounded and the increments decrease to zero. Specifically, the sequence oscillates around a value and the oscillations diminish as $n$ increases.

The sequence $\{a_n\}$ converges to a non-zero number. This is because the alternating series of powers of 2 converges, and the initial term $a_1 = 1$ ensures that the limit is non-zero.

### Analysis of $\{b_n\}$:

The sequence $\{b_n\}$ is defined as:
$$ b_n = \frac{2a_{n+1} - a_n}{a_n} $$

Substituting the recursive formula for $a_{n+1}$:
$$ b_n = \frac{2(a_n + (-1)^n 2^{-n}) - a_n}{a_n} = \frac{a_n + 2(-1)^n 2^{-n}}{a_n} = 1 + \frac{2(-1)^n 2^{-n}}{a_n} $$

As $n$ increases, the term $\frac{2(-1)^n 2^{-n}}{a_n}$ approaches zero because $2^{-n}$ diminishes rapidly and $a_n$ converges to a non-zero limit. Therefore, $\{b_n\}$ converges to 1, making it a Cauchy sequence.

### Conclusion:

- **Option 1** is correct because $\{a_n\}$ converges to a non-zero number, and $\{b_n\}$ is a Cauchy sequence, as shown by the analysis above.

### Why Other Options are Incorrect:

- **Option 2**: Incorrect because $\{a_n\}$ does not converge to zero; it converges to a non-zero number.
- **Option 3**: Incorrect because $\{b_n\}$ is a convergent sequence, not a non-convergent one.
- **Option 4**: Incorrect because $\{a_n\}$ does not converge to zero, and $\{b_n\}$ is a convergent sequence.

Thus, the correct answer is indeed Option 1, as it accurately describes the behavior of both sequences.