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Find the 11th term of the sequence

3, -6, 12, -24,...

The sequence provided is: \( 3, -6, 12, -24, \dots \).

This appears to be a geometric sequence where each term alternates in sign and is multiplied by a constant factor to get the next term. Let's confirm the pattern:

- From 3 to -6, the multiplication factor is \( -2 \).
- From -6 to 12, the multiplication factor is again \( -2 \).
- From 12 to -24, the factor is \( -2 \).

Thus, this is a geometric sequence with the first term \( a = 3 \) and common ratio \( r = -2 \).

The formula for the \( n \)-th term of a geometric sequence is:

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

For the 11th term (\( n = 11 \)):

\[
T_{11} = 3 \cdot (-2)^{11-1} = 3 \cdot (-2)^{10}
\]

Now, let me calculate \( (-2)^{10} \) and the 11th term.The 11th term of the sequence is \( 3072 \).

Would you like more details or have any further questions?

Here are some related questions to deepen your understanding:
1. How would you find the sum of the first 11 terms of this geometric sequence?
2. What happens if the common ratio is a fraction instead of an integer?
3. Can you derive the general formula for the sum of an infinite geometric series?
4. How would you determine the 20th term of this sequence?
5. What are the conditions for a geometric sequence to converge?

**Tip:** When working with geometric sequences, always check the common ratio between consecutive terms—it helps in quickly identifying the sequence pattern.

Find the 11th term of the sequence 3, -6, 12, -24,...

Learn how to find the 11th term of a geometric sequence with alternating signs. Using the formula T_n = a * r^(n-1), we calculate the 11th term, applying the common ratio -2. Suitable for students in Grades 9-12.

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