The sequence given is $8, 4, 2, \dots$, which is a geometric sequence.

Step 1: Identify the first term $a$ and the common ratio $r$

• First term $a = 8$
• The common ratio $r = \frac{4}{8} = \frac{1}{2}$

Step 2: Use the formula for the sum of the first $n$ terms of a geometric sequence:

$S_n = \frac{a(1 - r^n)}{1 - r}$ Where:

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

For the first 10 terms, we have:

• $a = 8$,
• $r = \frac{1}{2}$,
• $n = 10$.

Step 3: Substitute the values into the formula:

$S_{10} = \frac{8(1 - \left(\frac{1}{2}\right)^{10})}{1 - \frac{1}{2}}$

Step 4: Simplify the expression:

$S_{10} = \frac{8(1 - \left(\frac{1}{1024}\right))}{\frac{1}{2}} = \frac{8(1 - 0.0009765625)}{\frac{1}{2}} = \frac{8 \times 0.9990234375}{\frac{1}{2}} = 16 \times 0.9990234375$

$S_{10} \approx 15.984375$

Thus, the sum of the first 10 terms of the geometric sequence is approximately 15.98.

Would you like more details or have any questions?

Here are 5 related questions to further explore the concept:

1. What happens to the sum if the common ratio is greater than 1?
2. How do you find the sum of an infinite geometric sequence?
3. What is the sum of the first 5 terms of this sequence?
4. How can you determine whether a sequence is geometric?
5. What is the formula for the nth term of a geometric sequence?

Tip: The sum of an infinite geometric series only converges if the absolute value of the common ratio is less than 1.