Find the sum of the first 7 terms of the geometric sequence 3,-6,12,-24,48

To find the sum of the first 7 terms of the geometric sequence, we need to use the formula for the sum of the first \( n \) terms of a geometric sequence:

\[
S_n = a \frac{r^n - 1}{r - 1}
\]

where:
- \( S_n \) is the sum of the first \( n \) terms,
- \( a \) is the first term of the sequence,
- \( r \) is the common ratio, and
- \( n \) is the number of terms.

In this sequence:
- \( a = 3 \) (the first term),
- \( r = \frac{-6}{3} = -2 \) (the common ratio),
- \( n = 7 \) (the number of terms).

Now, substitute these values into the formula:

\[
S_7 = 3 \frac{(-2)^7 - 1}{-2 - 1}
\]

First, calculate \( (-2)^7 \):

\[
(-2)^7 = -128
\]

Then, calculate the sum:

\[
S_7 = 3 \frac{-128 - 1}{-3} = 3 \frac{-129}{-3} = 3 \times 43 = 129
\]

So, the sum of the first 7 terms of the sequence is \( 129 \).

Would you like more details or have any questions?

Here are some related questions to explore:
1. What is the sum of the first 10 terms of the same sequence?
2. How would the sum formula change if the sequence were arithmetic instead of geometric?
3. Can you derive the sum of an infinite geometric series if \( |r| < 1 \)?
4. What happens to the sum formula if \( r = 1 \)?
5. How does the sum change if the common ratio is positive?

**Tip:** In a geometric sequence, the common ratio is found by dividing any term by its preceding term.

Learn how to calculate the sum of the first 7 terms of the geometric sequence 3, -6, 12, -24, 48. Follow a step-by-step solution using the formula for the sum of a geometric series. Suitable for high school students in Grades 10-12.

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