Question

what is the \\(n^{\text{th}}\\) term rule for this arithmetic sequence?

the \\(n^{\text{th}}\\) term rule is \\(\square n - \square\\)

3, 12, 21, 30, ...

Explanation:

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<reasoning>

Identify the sequence terms

The given arithmetic sequence is:
\[ 3, 12, 21, 30, \dots \]
The first term is \( a_1 = 3 \).
The second term is \( a_2 = 12 \).
The third term is \( a_3 = 21 \).
The fourth term is \( a_4 = 30 \).

Find the common difference

An arithmetic sequence changes by a constant amount between consecutive terms. Let's calculate the common difference \( d \):
\[ d = a_2 - a_1 = 12 - 3 = 9 \]
\[ d = a_3 - a_2 = 21 - 12 = 9 \]
\[ d = a_4 - a_3 = 30 - 21 = 9 \]
The common difference is \( d = 9 \).

Determine the general nth term formula

The general formula for the \( n^{\text{th}} \) term of an arithmetic sequence is:
\[ a_n = a_1 + (n - 1)d \]
Substitute the known values \( a_1 = 3 \) and \( d = 9 \):
\[ a_n = 3 + (n - 1)9 \]

Simplify the expression

Expand and simplify the formula to match the required format \( \square n - \square \):
\[ a_n = 3 + 9n - 9 \]
\[ a_n = 9n - 6 \]

Comparing this to the template:
\[ \text{The } n^{\text{th}} \text{ term rule is } \square n - \square \]
We find that the first box contains \( 9 \) and the second box contains \( 6 \).
</reasoning>

<answer>
What is the \( n^{\text{th}} \) term rule for this arithmetic sequence?

The \( n^{\text{th}} \) term rule is <blank>9</blank>\(n -\) <blank>6</blank>
</answer>

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Answer:

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<reasoning>

Identify the sequence terms

The given arithmetic sequence is:
\[ 3, 12, 21, 30, \dots \]
The first term is \( a_1 = 3 \).
The second term is \( a_2 = 12 \).
The third term is \( a_3 = 21 \).
The fourth term is \( a_4 = 30 \).

Find the common difference

An arithmetic sequence changes by a constant amount between consecutive terms. Let's calculate the common difference \( d \):
\[ d = a_2 - a_1 = 12 - 3 = 9 \]
\[ d = a_3 - a_2 = 21 - 12 = 9 \]
\[ d = a_4 - a_3 = 30 - 21 = 9 \]
The common difference is \( d = 9 \).

Determine the general nth term formula

The general formula for the \( n^{\text{th}} \) term of an arithmetic sequence is:
\[ a_n = a_1 + (n - 1)d \]
Substitute the known values \( a_1 = 3 \) and \( d = 9 \):
\[ a_n = 3 + (n - 1)9 \]

Simplify the expression

Expand and simplify the formula to match the required format \( \square n - \square \):
\[ a_n = 3 + 9n - 9 \]
\[ a_n = 9n - 6 \]

Comparing this to the template:
\[ \text{The } n^{\text{th}} \text{ term rule is } \square n - \square \]
We find that the first box contains \( 9 \) and the second box contains \( 6 \).
</reasoning>

<answer>
What is the \( n^{\text{th}} \) term rule for this arithmetic sequence?

The \( n^{\text{th}} \) term rule is <blank>9</blank>\(n -\) <blank>6</blank>
</answer>

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