Write an explicit formula for a, start subscript, n, end subscript*an*​, the n, start superscript, th, end superscript*n*thterm of the sequence 2, comma, minus, 2, comma, minus, 6, comma, point, point, .2,−2,−6,....

The given sequence is:  
\[ 2, -2, -6, \dots \]

To find an explicit formula for the \(n\)-th term, observe the following pattern in the sequence:

1. The sequence starts with \(2\), followed by \(-2\), \(-6\), and so on.
2. The terms alternate in sign.
3. The absolute values of the terms are increasing by 4 each time: \(2, 2+4 = 6, 6+4 = 10\), etc.

### Step 1: Identify the general form
We can express the absolute value of the sequence's \(n\)-th term as a linear relationship:

\[
|a_n| = 2 + (n-1) \cdot 4
\]

This simplifies to:

\[
|a_n| = 4n - 2
\]

### Step 2: Account for alternating signs
The signs alternate in the pattern \(+, -, -, +, \dots\). The alternating signs can be expressed using \((-1)^{n+1}\). Therefore, the full formula for the \(n\)-th term of the sequence is:

\[
a_n = (-1)^{n+1} \cdot (4n - 2)
\]

### Final Formula:
The explicit formula for the \(n\)-th term is:

\[
a_n = (-1)^{n+1} \cdot (4n - 2)
\]

Would you like any further explanation on how this formula was derived, or any additional examples?

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Here are 5 related questions to deepen your understanding:

1. What is the \(5\)-th term of the sequence using the formula \(a_n = (-1)^{n+1}(4n-2)\)?
2. How can you determine if a sequence is arithmetic or geometric based on its terms?
3. What role does the alternating sign play in sequences, and how can it be recognized?
4. Can you derive the explicit formula for a similar sequence: \(3, -3, -9, \dots\)?
5. What changes in the formula if the sequence alternates in a different pattern, such as \(+, +, -, -, +, +, -\)?

**Tip**: Always look for patterns in both the signs and the magnitude of terms when analyzing a sequence. This helps in forming a general term formula efficiently.

Write an explicit formula for the nth term of the sequence 2, -2, -6, ...

Learn how to write the explicit formula for the nth term of the sequence 2, -2, -6, ... This solution explores key concepts like arithmetic progressions and alternating signs, with a step-by-step explanation of the formula an = (-1)^(n+1) * (4n - 2). Suitable for students in Grades 9-11.

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