what is the sum of 6th terms 3, 6, 12, 24, 48, 96 use geometric series

The sequence you provided is: \( 3, 6, 12, 24, 48, 96 \). This is a geometric series where each term is multiplied by 2 to get the next term. The common ratio \( r \) is 2, and the first term \( a \) is 3.

To find the sum of the first \( n \) terms of a geometric series, the formula is:

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

For this problem:
- \( a = 3 \)
- \( r = 2 \)
- \( n = 6 \)

Now, substitute these values into the formula:

\[
S_6 = 3 \cdot \frac{1 - 2^6}{1 - 2}
\]

Simplifying:

\[
S_6 = 3 \cdot \frac{1 - 64}{1 - 2} = 3 \cdot \frac{-63}{-1} = 3 \cdot 63 = 189
\]

Thus, the sum of the first six terms is \( S_6 = 189 \).

Would you like more details, or do you have any questions?

Here are 5 related questions to expand your understanding:
1. How do you derive the formula for the sum of a geometric series?
2. What happens to the sum if the common ratio is less than 1?
3. How do you find the sum of an infinite geometric series?
4. What is the difference between arithmetic and geometric series?
5. Can a geometric series have a negative common ratio?

**Tip:** In a geometric series, when the common ratio \( r \) is between -1 and 1, the series can converge to a finite sum, even if it has infinitely many terms.

Find the sum of the first 6 terms of the geometric series 3, 6, 12, 24, 48, 96 using the formula for the sum of a geometric series. Learn key concepts related to sequences and series, suitable for students in Grades 8-10.

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