Question
Find a general function for the sequence: 7,−14,21,−28,35,−42,…
an=−7n
an=(−7)n−1n
an=(−1)n(7n)
an=(−1)n7(n−1)
an=(−1)n−1(7n)
an=−7n
an=(−7)n−1n
an=(−1)n(7n)
an=(−1)n7(n−1)
an=(−1)n−1(7n)
Answers
GPT-4o mini
To analyze the sequence \(7, -14, 21, -28, 35, -42, \ldots\), let's look for a pattern.
1. The absolute values of the terms appear to follow the sequence \(7, 14, 21, 28, 35, 42, \ldots\), which can be expressed as:
\[
7n \quad \text{for } n = 1, 2, 3, \ldots
\]
This means that we can represent the magnitude of each term as \(7n\).
2. The signs of the sequence alternate starting with a positive value for the first term. We can represent this sign alternation using \((-1)^{n-1}\) or \((-1)^n\):
\[
\text{The first term is positive, so } (-1)^{n-1} \text{ would be suitable.}
\]
Combining these two observations, we can express the \(n\)-th term of the sequence as:
\[
a_n = (-1)^{n-1} (7n)
\]
To match it with the options provided, the correct form among the choices is:
\[
a_n = (-1)^{n-1} 7n
\]
Thus, the general function for the sequence is:
\[
\boxed{a_n = (-1)^{n-1} 7n}
\]
1. The absolute values of the terms appear to follow the sequence \(7, 14, 21, 28, 35, 42, \ldots\), which can be expressed as:
\[
7n \quad \text{for } n = 1, 2, 3, \ldots
\]
This means that we can represent the magnitude of each term as \(7n\).
2. The signs of the sequence alternate starting with a positive value for the first term. We can represent this sign alternation using \((-1)^{n-1}\) or \((-1)^n\):
\[
\text{The first term is positive, so } (-1)^{n-1} \text{ would be suitable.}
\]
Combining these two observations, we can express the \(n\)-th term of the sequence as:
\[
a_n = (-1)^{n-1} (7n)
\]
To match it with the options provided, the correct form among the choices is:
\[
a_n = (-1)^{n-1} 7n
\]
Thus, the general function for the sequence is:
\[
\boxed{a_n = (-1)^{n-1} 7n}
\]