To find the general term $a_n$ of the sequence:

The sequence is:
$\frac{10}{4}, \frac{10}{16}, \frac{10}{64}, \frac{10}{256}, \frac{10}{1024}, \dots$

Step 1: Analyze the Denominators

The denominators follow a pattern: $4, 16, 64, 256, 1024, \dots$ These values are powers of 4: $4 = 4^1, \quad 16 = 4^2, \quad 64 = 4^3, \quad 256 = 4^4, \quad 1024 = 4^5$

So, the denominator of the $n$-th term is $4^n$.

Step 2: Write the General Term

Each term in the sequence has the form: $a_n = \frac{10}{4^n}$

Thus, the general term of the sequence is: $a_n = \frac{10}{4^n}$

Conclusion:

The general formula for the $n$-th term is: $a_n = \frac{10}{4^n}$

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Do you want more details or have any questions?

Here are 5 related questions to explore further:

1. How would the formula change if the sequence began with $20$ instead of $10$?
2. Can you find the sum of the infinite series $\frac{10}{4^n}$?
3. What would the general term be if the denominators followed powers of 3 instead?
4. How can you verify that this formula works for any given term of the sequence?
5. How would the pattern change if the numerators followed a pattern like $10, 20, 40, \dots$?

Tip: When recognizing a pattern, always look for powers or multiplicative relationships in both the numerator and denominator.