Find a formula for the general term a^n of the sequence assuming the pattern of the first few terms continues. [10/4, 10/16, 10/64, 10/256, 10/1024]

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.

Explore how to find the general term of the sequence [10/4, 10/16, 10/64, 10/256, 10/1024] by identifying exponential patterns in the denominators. Learn the step-by-step solution suitable for Grades 9-12.

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