Question

What is a formula for the nth term of the given sequence?
8, comma, minus, 4, comma, 2, point, point, .
8,−4,2...
Answer
Multiple Choice Answers
a, start subscript, n, end subscript, equals, minus, 16, left parenthesis, minus, one half, right parenthesis, start superscript, minus, n, end superscripta
n
​
=−16(−
2
1
​
)
−n

a, start subscript, n, end subscript, equals, minus, 16, left parenthesis, minus, one half, right parenthesis, start superscript, n, minus, 1, end superscripta
n
​
=−16(−
2
1
​
)
n−1

a, start subscript, n, end subscript, equals, 8, left parenthesis, minus, one half, right parenthesis, start superscript, n, minus, 1, end superscripta
n
​
=8(−
2
1
​
)
n−1

a, start subscript, n, end subscript, equals, 8, left parenthesis, minus, 2, right parenthesis, to the power na
n
​
=8(−2)
n

Answer 1

To find a formula for the \( n \)-th term of the sequence 8, -4, 2..., we first observe the pattern of the terms.

1. The first term \( a_1 = 8 \).
2. The second term \( a_2 = -4 \).
3. The third term \( a_3 = 2 \).

To identify a pattern, we can examine how the terms change:
- Going from 8 to -4: \( -4 = 8 \times (-\frac{1}{2}) \)
- Going from -4 to 2: \( 2 = -4 \times (-\frac{1}{2}) \)

It appears that each term is multiplied by \( -\frac{1}{2} \) to get the next term. This indicates that the sequence can be expressed recursively:

\[
a_n = -\frac{1}{2} a_{n-1}
\]

Following this pattern, we can establish a closed-form expression for the \( n \)-th term. The \( n \)-th term can be represented as:

\[
a_n = 8 \left(-\frac{1}{2}\right)^{n-1}
\]

Thus, the correct formula for the \( n \)-th term of the sequence is:

\[
a_n = 8 \left(-\frac{1}{2}\right)^{n-1}
\]

Based on the multiple choice answers you provided, this matches:

- **Answer**: \( a_n = 8 \left(-\frac{1}{2}\right)^{n-1} \)

So, the correct selection from your options is:

**c)** \( a_n = 8 \left(-\frac{1}{2}\right)^{n-1} \)