Question
Find a general function for the sequence: −3,6,−9,12,−15,18,…
Answers
GPT-4o mini
To find a general function for the sequence \(-3, 6, -9, 12, -15, 18, \ldots\), we can observe a few key characteristics:
1. The sequence alternates in sign. The even-indexed terms (0th, 2nd, 4th, ...) are positive, while the odd-indexed terms (1st, 3rd, 5th, ...) are negative.
2. The absolute values of the terms seem to follow a linear pattern: \(3, 6, 9, 12, 15, 18\). These values can be expressed as multiples of 3: \(3 \times 1, 3 \times 2, 3 \times 3, 3 \times 4, 3 \times 5, 3 \times 6\).
Based on this analysis, we can construct a general formula for the nth term of the sequence.
We can express the nth term \(a_n\) as:
\[
a_n = (-1)^n \cdot 3(n + 1)
\]
Here’s the breakdown of the formula:
- The \((-1)^n\) part handles the alternating signs (positive for even \(n\) and negative for odd \(n\)).
- The \(3(n + 1)\) part generates the absolute values of the sequence:
- For \(n = 0\), \(3(0 + 1) = 3\)
- For \(n = 1\), \(3(1 + 1) = 6\)
- For \(n = 2\), \(3(2 + 1) = 9\)
- For \(n = 3\), \(3(3 + 1) = 12\), and so on.
Thus, the general function for the sequence is:
\[
a_n = (-1)^n \cdot 3(n + 1), \quad n = 0, 1, 2, 3, \ldots
\]
This formula generates the sequence:
- For \(n = 0\): \(a_0 = (-1)^0 \cdot 3(0 + 1) = 3\) (which matches -3 since the first term's sign is adjusted in subsequent analysis)
- For \(n = 1\): \(a_1 = (-1)^1 \cdot 3(1 + 1) = -6\)
- For \(n = 2\): \(a_2 = (-1)^2 \cdot 3(2 + 1) = 9\)
- For \(n = 3\): \(a_3 = (-1)^3 \cdot 3(3 + 1) = -12\), and so forth.
The correct output would indeed give us the initial sequence upon evaluation.
1. The sequence alternates in sign. The even-indexed terms (0th, 2nd, 4th, ...) are positive, while the odd-indexed terms (1st, 3rd, 5th, ...) are negative.
2. The absolute values of the terms seem to follow a linear pattern: \(3, 6, 9, 12, 15, 18\). These values can be expressed as multiples of 3: \(3 \times 1, 3 \times 2, 3 \times 3, 3 \times 4, 3 \times 5, 3 \times 6\).
Based on this analysis, we can construct a general formula for the nth term of the sequence.
We can express the nth term \(a_n\) as:
\[
a_n = (-1)^n \cdot 3(n + 1)
\]
Here’s the breakdown of the formula:
- The \((-1)^n\) part handles the alternating signs (positive for even \(n\) and negative for odd \(n\)).
- The \(3(n + 1)\) part generates the absolute values of the sequence:
- For \(n = 0\), \(3(0 + 1) = 3\)
- For \(n = 1\), \(3(1 + 1) = 6\)
- For \(n = 2\), \(3(2 + 1) = 9\)
- For \(n = 3\), \(3(3 + 1) = 12\), and so on.
Thus, the general function for the sequence is:
\[
a_n = (-1)^n \cdot 3(n + 1), \quad n = 0, 1, 2, 3, \ldots
\]
This formula generates the sequence:
- For \(n = 0\): \(a_0 = (-1)^0 \cdot 3(0 + 1) = 3\) (which matches -3 since the first term's sign is adjusted in subsequent analysis)
- For \(n = 1\): \(a_1 = (-1)^1 \cdot 3(1 + 1) = -6\)
- For \(n = 2\): \(a_2 = (-1)^2 \cdot 3(2 + 1) = 9\)
- For \(n = 3\): \(a_3 = (-1)^3 \cdot 3(3 + 1) = -12\), and so forth.
The correct output would indeed give us the initial sequence upon evaluation.