To find the sum of the first 10 terms of a geometric sequence, you can use the formula:

$S_n = a \frac{1 - r^n}{1 - r}$

Where:

• $S_n$ is the sum of the first $n$ terms,
• $a$ is the first term,
• $r$ is the common ratio, and
• $n$ is the number of terms.

Given the sequence $4, 2, 1, \dots$:

• $a = 4$
• $r = \frac{2}{4} = \frac{1}{2}$
• $n = 10$

Plugging these values into the formula:

$S_{10} = 4 \cdot \frac{1 - \left(\frac{1}{2}\right)^{10}}{1 - \frac{1}{2}}$

Simplify the expression:

$S_{10} = 4 \cdot \frac{1 - \left(\frac{1}{2}\right)^{10}}{\frac{1}{2}} = 4 \cdot 2 \cdot \left(1 - \frac{1}{1024}\right)$

$S_{10} = 8 \cdot \left(1 - \frac{1}{1024}\right) = 8 \cdot \frac{1023}{1024} = \frac{8184}{1024} = 7.9921875$

So, the sum of the first 10 terms is approximately $7.992$.

Would you like further details or have any questions?

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Here are some related questions you might find useful:

1. What is the sum of the first 15 terms of the same sequence?
2. How do you find the sum of an infinite geometric series?
3. What happens if the common ratio $r$ is greater than 1?
4. Can you derive the formula for the sum of a finite geometric series?
5. What is the sum of the first 10 terms if the sequence started at 8 instead of 4?

Tip: Always check if the common ratio $r$ is between -1 and 1 when calculating the sum of an infinite geometric series.