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Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm. 20 , 60 , 1...Question
Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.
2 , 8 , 32 , ...
Write your answer using decimals and integers.
an =
(
)n - 1
2 , 8 , 32 , ...
Write your answer using decimals and integers.
an =
(
)n - 1
Answers
Answer
write the answer using decimals and intergers
Answer
Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.
4 , –12 , 36 , ...
Write your answer using decimals and integers.
an =
(
)n - 1
4 , –12 , 36 , ...
Write your answer using decimals and integers.
an =
(
)n - 1
Answer
Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.
1 , 3 , 9 , ...
Write your answer using decimals and integers.
an =
(
)n - 1
1 , 3 , 9 , ...
Write your answer using decimals and integers.
an =
(
)n - 1
Answer
write using decimals and intergers
Answered by
GPT-4o mini
To find the equation that describes the given sequence \(2, 8, 32, \ldots\), let's first analyze the pattern in the terms.
1. The first term (for \(n = 1\)) is \(2\).
2. The second term (for \(n = 2\)) is \(8\).
3. The third term (for \(n = 3\)) is \(32\).
We notice that:
- \(2 = 2 \cdot 2^0\)
- \(8 = 2 \cdot 2^2\)
- \(32 = 2 \cdot 2^4\)
In each case, we can rewrite the terms as:
- \(a_1 = 2 \cdot 2^{(n-1) \cdot 2}\)
From this, we can determine the formula for the \(n\)th term as follows:
\[
a_n = 2 \cdot 2^{(n-1) \cdot 2}
\]
This simplifies to:
\[
a_n = 2^{2n - 1}
\]
Thus, we can write the equation for the \(n\)th term of the sequence as:
\[
a_n = 2^{2n - 1}
\]
So, the final answer is:
\[
a_n = 2^{2n - 1}
\]
1. The first term (for \(n = 1\)) is \(2\).
2. The second term (for \(n = 2\)) is \(8\).
3. The third term (for \(n = 3\)) is \(32\).
We notice that:
- \(2 = 2 \cdot 2^0\)
- \(8 = 2 \cdot 2^2\)
- \(32 = 2 \cdot 2^4\)
In each case, we can rewrite the terms as:
- \(a_1 = 2 \cdot 2^{(n-1) \cdot 2}\)
From this, we can determine the formula for the \(n\)th term as follows:
\[
a_n = 2 \cdot 2^{(n-1) \cdot 2}
\]
This simplifies to:
\[
a_n = 2^{2n - 1}
\]
Thus, we can write the equation for the \(n\)th term of the sequence as:
\[
a_n = 2^{2n - 1}
\]
So, the final answer is:
\[
a_n = 2^{2n - 1}
\]
Answered by
GPT-4o mini
To express the formula for the sequence \(2, 8, 32, \ldots\) using only integers and decimals, we can rewrite it in a clearer form.
The nth term of the sequence can be described using powers of 2. From the previous analysis, we found that:
\[
a_n = 2^{2n - 1}
\]
However, to align with your request which involves the specific formatting and mention of the sequence requirement, the formula can be presented as:
\[
a_n = 2 \cdot 4^{n - 1}
\]
where \(4\) can also be expressed as \(2^2\), and it uses integers.
In summary, the equation can be rewritten as:
\[
a_n = 2 \cdot 4^{n - 1}
\]
This encapsulates the sequence using integers and the integer exponent for \(4^{n - 1}\).
The nth term of the sequence can be described using powers of 2. From the previous analysis, we found that:
\[
a_n = 2^{2n - 1}
\]
However, to align with your request which involves the specific formatting and mention of the sequence requirement, the formula can be presented as:
\[
a_n = 2 \cdot 4^{n - 1}
\]
where \(4\) can also be expressed as \(2^2\), and it uses integers.
In summary, the equation can be rewritten as:
\[
a_n = 2 \cdot 4^{n - 1}
\]
This encapsulates the sequence using integers and the integer exponent for \(4^{n - 1}\).
Answered by
GPT-4o mini
To find the equation that describes the sequence \(4, -12, 36, \ldots\), let's analyze the terms.
1. The first term (\(n = 1\)) is \(4\).
2. The second term (\(n = 2\)) is \(-12\).
3. The third term (\(n = 3\)) is \(36\).
Now, let's look for a pattern in the terms:
1. For \(n = 1\): \(4\)
2. For \(n = 2\): \(-12\) (which is \(4 \cdot -3\))
3. For \(n = 3\): \(36\) (which is \(4 \cdot 9\))
It seems like we can extract some pattern from the coefficients when observing:
- The factor seems to alternate in sign and increase in magnitude.
To analyze this mathematically, let's express this as:
- The signs alternate, suggesting a factor of \((-1)^{n-1}\).
- The absolute values of the terms seem to follow the pattern of \(4 \cdot 3^{n-1}\) since \(-12\) is \(-4 \cdot 3\) (when \(n=2\)) and \(36\) is \(4 \cdot 9\) (when \(n=3\)).
Combining these insights, we can state:
\[
a_n = 4 \cdot (-3)^{n-1}
\]
Thus, the final answer as per your request is:
\[
a_n = 4 \cdot (-3)^{n - 1}
\]
1. The first term (\(n = 1\)) is \(4\).
2. The second term (\(n = 2\)) is \(-12\).
3. The third term (\(n = 3\)) is \(36\).
Now, let's look for a pattern in the terms:
1. For \(n = 1\): \(4\)
2. For \(n = 2\): \(-12\) (which is \(4 \cdot -3\))
3. For \(n = 3\): \(36\) (which is \(4 \cdot 9\))
It seems like we can extract some pattern from the coefficients when observing:
- The factor seems to alternate in sign and increase in magnitude.
To analyze this mathematically, let's express this as:
- The signs alternate, suggesting a factor of \((-1)^{n-1}\).
- The absolute values of the terms seem to follow the pattern of \(4 \cdot 3^{n-1}\) since \(-12\) is \(-4 \cdot 3\) (when \(n=2\)) and \(36\) is \(4 \cdot 9\) (when \(n=3\)).
Combining these insights, we can state:
\[
a_n = 4 \cdot (-3)^{n-1}
\]
Thus, the final answer as per your request is:
\[
a_n = 4 \cdot (-3)^{n - 1}
\]
Answered by
GPT-4o mini
To find an equation that describes the sequence \(1, 3, 9, \ldots\), let's first identify the pattern in the terms:
1. The first term (\(n = 1\)) is \(1\).
2. The second term (\(n = 2\)) is \(3\).
3. The third term (\(n = 3\)) is \(9\).
We can observe that:
- The first term can be expressed as \(3^0 = 1\),
- The second term can be expressed as \(3^1 = 3\),
- The third term can be expressed as \(3^2 = 9\).
From this pattern, we see that the nth term of the sequence can be represented as:
\[
a_n = 3^{n-1}
\]
So, the final answer is:
\[
a_n = 3^{n - 1}
\]
1. The first term (\(n = 1\)) is \(1\).
2. The second term (\(n = 2\)) is \(3\).
3. The third term (\(n = 3\)) is \(9\).
We can observe that:
- The first term can be expressed as \(3^0 = 1\),
- The second term can be expressed as \(3^1 = 3\),
- The third term can be expressed as \(3^2 = 9\).
From this pattern, we see that the nth term of the sequence can be represented as:
\[
a_n = 3^{n-1}
\]
So, the final answer is:
\[
a_n = 3^{n - 1}
\]
Answered by
GPT-4o mini
To write the equation for the sequence \(1, 3, 9, \ldots\) using decimals and integers, we can express the equation in a slightly different form that adheres to your request.
The equation I've provided is already in a simple integer and exponent form, but we can express powers in a different way. We can rewrite \(3^{n-1}\) in a form that reflects the structure you've requested, which is:
\[
a_n = 3 \cdot (1)^{n - 1}
\]
This is because \(3 \cdot (1)^{n - 1}\) is mathematically equivalent to \(3^{n-1}\) when considering \(n = 1\) leading to \(1\) (which keeps the value consistent but may seem redundant).
However, if the preference is strictly for a specific format, the simplest and most standard version remains:
\[
a_n = 3^{n - 1}
\]
There isn't a meaningful use of decimals in this context that would apply. Thus, to reiterate using just integers and a spatial layout:
\[
a_n = 3^{n - 1}
\]
is indeed the simplest expression, as \(3\) and \(1\) in the context of exponents are integers. If you need a specific format or component without using exponents, please clarify!
The equation I've provided is already in a simple integer and exponent form, but we can express powers in a different way. We can rewrite \(3^{n-1}\) in a form that reflects the structure you've requested, which is:
\[
a_n = 3 \cdot (1)^{n - 1}
\]
This is because \(3 \cdot (1)^{n - 1}\) is mathematically equivalent to \(3^{n-1}\) when considering \(n = 1\) leading to \(1\) (which keeps the value consistent but may seem redundant).
However, if the preference is strictly for a specific format, the simplest and most standard version remains:
\[
a_n = 3^{n - 1}
\]
There isn't a meaningful use of decimals in this context that would apply. Thus, to reiterate using just integers and a spatial layout:
\[
a_n = 3^{n - 1}
\]
is indeed the simplest expression, as \(3\) and \(1\) in the context of exponents are integers. If you need a specific format or component without using exponents, please clarify!