This problem involves finding the sum of the first 8 terms of a finite geometric sequence. Let’s solve it step by step:

Step 1: Recognize the terms of the geometric sequence

The sequence is:
$24, -48, 96, \dots$

The first term ($a_1$) is: $a_1 = 24$

The common ratio ($r$) can be calculated by dividing the second term by the first term: $r = \frac{-48}{24} = -2$

Step 2: Use the sum formula for a finite geometric series

The formula for the sum of the first $n$ terms of a geometric sequence is: $S_n = \frac{a_1(1 - r^n)}{1 - r}$

Here:

• $a_1 = 24$,
• $r = -2$,
• $n = 8$.

Step 3: Substitute values into the formula

$S_8 = \frac{24(1 - (-2)^8)}{1 - (-2)}$

First, calculate $(-2)^8$: $(-2)^8 = 256$

Substitute this into the formula: $S_8 = \frac{24(1 - 256)}{1 + 2}$

Simplify the numerator: $1 - 256 = -255$

So: $S_8 = \frac{24(-255)}{3}$

Step 4: Simplify the fraction

$24 \times (-255) = -6120$

$S_8 = \frac{-6120}{3} = -2040$

Final Answer:

The sum of the first 8 terms of the sequence is: $\boxed{-2040}$

Would you like a breakdown of the concepts or have additional questions?

Related Questions:

1. What happens to the sum as $n \to \infty$ if $|r| < 1$?
2. How do you find the sum of an infinite geometric series?
3. How do you determine if a sequence is geometric from its terms?
4. What happens if the common ratio $r$ is a fraction instead of an integer?
5. Can you derive the formula for the sum of a finite geometric series?

Tip:

Always check the absolute value of $r$. If $|r| > 1$, the terms grow rapidly, and the sum of the series increases significantly with each additional term.