To determine the general formula for the sequence $\frac{1}{6}, \frac{1}{5}, \frac{6}{25}$, let's first analyze the terms and see if there's a pattern.

The sequence is: $a_1 = \frac{1}{6}, \quad a_2 = \frac{1}{5}, \quad a_3 = \frac{6}{25}$

Let's look at the numerators and denominators separately to uncover any possible pattern.

Numerators:

• For $a_1$, the numerator is 1.
• For $a_2$, the numerator is also 1.
• For $a_3$, the numerator is 6.

Denominators:

• For $a_1$, the denominator is 6.
• For $a_2$, the denominator is 5.
• For $a_3$, the denominator is 25.

Step 1: Identifying patterns

• Numerators: The numerators initially stay the same for the first two terms, and then jump to 6. This suggests that the numerator could be constant for some terms but then increase.
• Denominators: The sequence of denominators appears to decrease initially (6 to 5), and then there's a sudden jump to 25. This suggests that the denominator might follow some multiplicative or exponential pattern.

Step 2: Possible general formula

Given the jumps in the numerators and denominators, we hypothesize the following general form for the nth term: $a_n = \frac{ \text{Numerator Pattern} }{ \text{Denominator Pattern} }$

We can test this with some trial-and-error, but without more terms, it’s hard to deduce an exact formula directly from just three terms. If you have more terms, this could help narrow down the sequence.

Would you like to explore this further with additional terms or any other specific aspects? Feel free to share any more information about the sequence if available.

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Follow-up Questions:

1. Are there additional terms in the sequence you could provide?
2. Does the sequence have any real-world application or context?
3. Do the terms represent a known mathematical series (e.g., Fibonacci, geometric, etc.)?
4. What is the behavior or trend of the sequence as $n$ increases?
5. How might you verify the formula for $a_n$ with larger terms?

Tip:

Always look at the sequence's numerators and denominators separately first. This can often give you insight into separate patterns that may combine to form a general formula for the nth term.