设有理数数列 \(x_1, x_2, \dots\) 定义如下：\(x_1 = \frac{25}{...

Exploring recurrence

We’re given a recurrence for a sequence $x_1, x_2, \ldots$, with the first term $x_1 = \frac{25}{11}$ and the recurrence formula:

I’m trying to figure out how to handle this recurrence by possibly simplifying it with substitution. I wonder if it helps to introduce a new variable, like $y_k = \frac{1}{x_k}$, to gradually simplify it, or maybe find a telescoping pattern.

Let’s analyze further!

Analyzing recurrence transformations

I’m trying different transformations for the recurrence: maybe $x_k = \frac{a_k}{b_k}$, or considering similarities to cosine recurrences. I think the presence of $-1$ inside the recurrence makes it not exactly like a Chebyshev recurrence, though.

Let’s try defining $u_k = x_k + \frac{1}{x_k}$ and see if it helps simplify things, but that doesn’t seem to work either. I might consider solving for fixed points, which gives me potential limit values of $x = \frac{1}{2}$ or $-1$. Still need to figure out the exact terms for $x_{2025}$.

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Exploring transformations and recurrences

I’ve tried different transformations to handle the recurrence, looking for a simpler approach. Using substitutions like $y_k = \frac{x_k - 1}{x_k + 1}$ got us somewhere, as it turns the recurrence into a well-known form resembling a Chebyshev recurrence. I found $y_1 = \frac{7}{18}$, and with that, I computed successive terms in the sequence. But this quickly becomes messy. This recurrence likely connects with recursive structures, so let’s see if we can find a more direct approach or closed form.

Exploring recurrence patterns

I’m still trying to find a simpler way to handle the recurrence. Using the substitution $y_k = \frac{x_k - 1}{x_k + 1}$ led to a recurrence that resembles the cosine triple angle identity, but with a twist due to the coefficients. Now, I’m trying to check if the sequence is periodic by calculating the terms one after another. It looks like the sequence might oscillate after a bit, with $y_1 \approx 0.3889$ and $y_2 \approx -0.2731$. If it’s periodic, then I’ll try using mod-based tricks to find the exact term.

Investigating recurrence structure

I’m considering if the recurrence can be represented by a projective matrix, defining the sequence as $x_k = \frac{a_k}{b_k}$ with initial values $a_1 = 25$ and $b_1 = 11$. From this, I derive relationships for $a_{k+1}$ and $b_{k+1}$, and I notice patterns forming in terms of sums, $a_k + b_k = 36^{(2^{k-1})}$. I'll also need to track the greatest common divisor $d_k$, but the key now seems to be understanding how this recurrence impacts the power of 3 in the terms.

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