The given sequence is: 4, 2, 1, ...

This is a geometric sequence where each term after the first is found by multiplying the previous term by a constant ratio. To determine the sum of the first 10 terms, let's first identify the relevant values:

1. First term $a_1$: The first term is $4$.
2. Common ratio $r$: The ratio between the terms is $\frac{2}{4} = \frac{1}{2}$.

Formula for the sum of the first $n$ terms of a geometric sequence:

$S_n = \frac{a_1(1 - r^n)}{1 - r}, \quad \text{for} \quad r \neq 1$ Where:

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

Plugging in the values:

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

The sum of the first 10 terms: $S_{10} = \frac{4(1 - \left(\frac{1}{2}\right)^{10})}{1 - \frac{1}{2}}$

Simplifying:

1. $1 - \left(\frac{1}{2}\right)^{10} = 1 - \frac{1}{1024} = \frac{1023}{1024}$,
2. $1 - \frac{1}{2} = \frac{1}{2}$.

Now substitute these into the formula: $S_{10} = \frac{4 \cdot \frac{1023}{1024}}{\frac{1}{2}} = 4 \cdot \frac{1023}{1024} \cdot 2 = \frac{8 \cdot 1023}{1024} = \frac{8184}{1024} = 7.9921875$

Thus, the sum of the first 10 terms is approximately: $S_{10} \approx 7.99$

Would you like further details or clarifications?

Here are some related questions:

1. How do we determine the common ratio of a geometric sequence?
2. What is the sum formula for an arithmetic series?
3. How do you calculate the sum of infinite terms in a geometric series?
4. Can the sum formula be used if the ratio is greater than 1?
5. What happens if the ratio is negative in a geometric series?

Tip: Always check the ratio in geometric sequences to ensure the sum formula is applicable!