Question

The eleventh term of the sequence 3, -6, 12, -24, ..., is

Answer 1

To find the eleventh term of the sequence 3, -6, 12, -24, ..., we first need to identify the pattern in the sequence.

Looking at the sequence:

- The first term is \( a_1 = 3 \).
- The second term is \( a_2 = -6 \).
- The third term is \( a_3 = 12 \).
- The fourth term is \( a_4 = -24 \).

Let's examine how the terms relate to each other:

1. From \( a_1 \) to \( a_2 \):
\[
a_2 = a_1 \times -2 = 3 \times -2 = -6
\]

2. From \( a_2 \) to \( a_3 \):
\[
a_3 = a_2 \times -2 = -6 \times -2 = 12
\]

3. From \( a_3 \) to \( a_4 \):
\[
a_4 = a_3 \times -2 = 12 \times -2 = -24
\]

From these calculations, we see that each term can be obtained from the previous term by multiplying by \(-2\). Therefore, we can express the \( n \)-th term of the sequence using the formula:

\[
a_n = 3 \times (-2)^{n-1}
\]

Now, we can find the eleventh term, \( a_{11} \):

\[
a_{11} = 3 \times (-2)^{11-1} = 3 \times (-2)^{10}
\]

Calculating \( (-2)^{10} \):

\[
(-2)^{10} = 1024
\]

Thus, substituting back into the formula:

\[
a_{11} = 3 \times 1024 = 3072
\]

Therefore, the eleventh term of the sequence is

\[
\boxed{3072}
\]