What is the sum of the first 10 terms of the geometric sequence 4,2,1??

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.

Learn how to calculate the sum of the first 10 terms of the geometric sequence 4, 2, 1 using the formula for the sum of a geometric series. This problem involves understanding geometric sequences, common ratios, and applying the sum formula. Suitable for high school students.

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