Question

Given the sequence:

-2, 4, -8, 16, -32,...

What is the value of the 15th term?

(1 point)
Responses

65,536
65,536

-8,192
-8,192

-32,758
-32,758

16,384

Answer 1

To find the 15th term of the sequence \(-2, 4, -8, 16, -32, \ldots\), we can first identify the pattern in the sequence.

We see that:
- The sequence alternates in sign (negative and positive).
- The absolute values of the terms appear to follow a pattern of doubling in magnitude.

Examining the terms:
1. The first term: \(-2\) (which we can see as \(-2 \times 2^0 = -2\))
2. The second term: \(4\) (which can be written as \(2 \times 2^1 = 4\))
3. The third term: \(-8\) (which can be written as \(-2 \times 2^2 = -8\))
4. The fourth term: \(16\) (which can be written as \(2 \times 2^3 = 16\))
5. The fifth term: \(-32\) (which can be written as \(-2 \times 2^4 = -32\))

This sequence can be generalized as:

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

where \(n\) is the term number starting from 1.

To find the 15th term (\(a_{15}\)):
\[
a_{15} = (-1)^{15} \cdot 2^{15}
\]
Since \(15\) is odd, \((-1)^{15} = -1\). Thus:
\[
a_{15} = -1 \cdot 2^{15}
\]

Calculating \(2^{15}\):
\[
2^{15} = 32768
\]

Putting it together:
\[
a_{15} = -32768
\]

However, looking at the provided responses, it seems there is a need for clarification on values, but considering our calculations, the value of the 15th term is:

\(-32,768\).

Thus, the answer is:
**-32,768**