To find the sum of the first 10 terms of the geometric sequence 4, 2, 1, ..., we can use the formula for the sum of the first $n$ terms of a geometric sequence:

$S_n = \frac{a_1(1 - r^n)}{1 - r}$

Where:

• $S_n$ is the sum of the first $n$ terms,
• $a_1$ is the first term of the sequence,
• $r$ is the common ratio,
• $n$ is the number of terms.

For the given sequence:

• $a_1 = 4$
• $r = \frac{2}{4} = \frac{1}{2}$
• $n = 10$

Substitute these values into the formula:

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

Simplify the expression:

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

$S_{10} = \frac{4 \left(\frac{1023}{1024}\right)}{\frac{1}{2}} = 8 \times \frac{1023}{1024} = \frac{8196}{1024} = 7.9921875$

So, the sum of the first 10 terms of the geometric sequence is approximately 7.9921875.

Would you like more details or have any questions?

Here are 5 related questions:

1. How do you find the sum of an infinite geometric series?
2. What happens if the common ratio $r$ is greater than 1?
3. How do you determine if a series converges or diverges?
4. Can you derive the formula for the sum of a geometric series?
5. How do you calculate the sum of the first 20 terms for the same sequence?

Tip: When the common ratio $r$ is a fraction between 0 and 1, the terms of the sequence decrease, and the sum approaches a finite value as more terms are added.