Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm.

Question

2	, 	8	, 	32	, 	...

Write your answer using decimals and integers.

an = 
(
)n - 1

GPT-4o mini · Answer

To find the equation that describes the given sequence $2, 8, 32, \ldots$, let's first analyze the pattern in the terms.

The first term (for $n = 1$) is $2$.
The second term (for $n = 2$) is $8$.
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:

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

Answer

write the answer using decimals and intergers

GPT-4o mini · Answer

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 $4^{n - 1}$.

Answer

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm. 4 , –12 , 36 , ... Write your answer using decimals and integers. an = ( )n - 1

GPT-4o mini · Answer

To find the equation that describes the sequence $4, -12, 36, \ldots$, let's analyze the terms.

The first term ($n = 1$) is $4$.
The second term ($n = 2$) is $-12$.
The third term ($n = 3$) is $36$.

Now, let's look for a pattern in the terms:

For $n = 1$: $4$
For $n = 2$: $-12$ (which is $4 \cdot -3$)
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:

\[ a_n = 4 \cdot (-3)^{n - 1} \]

Answer

Writeanequationtodescribethesequencebelow.Usentorepresentthepositionofaterminthesequence,wheren = 1forthefirstterm. 1 , 3 , 9 , ... Write your answer using decimals and integers. an = ( )n - 1

GPT-4o mini · Answer

To find an equation that describes the sequence $1, 3, 9, \ldots$, let's first identify the pattern in the terms:

The first term ($n = 1$) is $1$.
The second term ($n = 2$) is $3$.
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:

\[ a_n = 3^{n - 1} \]

Answer

write using decimals and intergers

GPT-4o mini · Answer

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!