Given the problem:

$a_1 = 1, \quad (a_{n+1})^3 \times (a_n)^2 = 2, \quad a_n > 0$

We are asked to find the limit of the sequence $\{a_n\}$ as $n \to \infty$.

Solution:

1. Given Recurrence Relation: $(a_{n+1})^3 \times (a_n)^2 = 2$ We can rewrite this as: $(a_{n+1})^3 = \frac{2}{(a_n)^2}$ Taking the cube root of both sides: $a_{n+1} = \left(\frac{2}{(a_n)^2}\right)^{\frac{1}{3}} = \frac{2^{\frac{1}{3}}}{a_n^{\frac{2}{3}}}$

2. Assuming the sequence converges to a limit $L$, we can set $a_n \approx L$ as $n \to \infty$: $L = \frac{2^{\frac{1}{3}}}{L^{\frac{2}{3}}}$ To find $L$, multiply both sides by $L^{\frac{2}{3}}$: $L^{\frac{5}{3}} = 2^{\frac{1}{3}}$ Raise both sides to the power of $\frac{3}{5}$: $L = 2^{\frac{1}{5}}$

Conclusion:

The limit of the sequence $\{a_n\}$ as $n \to \infty$ is $L = 2^{\frac{1}{5}}$.

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Would you like more details or have any other questions?

Related Questions:

1. How do you prove the convergence of the sequence $\{a_n\}$?
2. What happens to the sequence if we change the initial condition $a_1$?
3. How does the recurrence relation change if the powers of $a_{n+1}$ and $a_n$ are different?
4. Can this method be used to find the limit of other types of recursive sequences?
5. What are the implications of $a_n > 0$ in the behavior of the sequence?

Tip:

When dealing with recursive sequences, assume the sequence converges and set up the equation involving the limit to solve for it. This is often the simplest way to find the limit of the sequence.