Question
Find a general function for the sequence: 8,−16,24,−32,40,−48,…
Answers
GPT-4o mini
To find a general function for the sequence \( 8, -16, 24, -32, 40, -48, \ldots \), let's first observe the pattern in the terms.
1. **Identifying the Position and Signs**:
- The sequence alternates signs: positive for even indexed terms (0, 2, 4, ...) and negative for odd indexed terms (1, 3, 5, ...).
- We can express the sign alternation using \((-1)^n\), where \(n\) is the term index starting from 0.
2. **Identifying the Sequence of Absolute Values**:
- The absolute values of the sequence are: \(8, 16, 24, 32, 40, 48, \ldots\).
- These values appear to increase by a consistent amount, i.e., the increments are \(8, 8, 8, 8, 8, \ldots\).
- Therefore, the absolute value can be expressed as a linear function of the index \(n\).
3. **Finding the Linear Function**:
- To find the pattern in the absolute values, we note that:
- At \(n = 0\): \(8 = 8(1)\)
- At \(n = 1\): \(16 = 8(2)\)
- At \(n = 2\): \(24 = 8(3)\)
- At \(n = 3\): \(32 = 8(4)\)
- At \(n = 4\): \(40 = 8(5)\)
- At \(n = 5\): \(48 = 8(6)\)
- We can see that the term can be represented as:
\[
8(n + 1)
\]
4. **Combining the Results**:
- Therefore, the \(n^{th}\) term of the sequence can be expressed as:
\[
a_n = (-1)^n \cdot 8(n + 1)
\]
Thus, the general function for the sequence is:
\[
a_n = (-1)^n \cdot 8(n + 1)
\]
1. **Identifying the Position and Signs**:
- The sequence alternates signs: positive for even indexed terms (0, 2, 4, ...) and negative for odd indexed terms (1, 3, 5, ...).
- We can express the sign alternation using \((-1)^n\), where \(n\) is the term index starting from 0.
2. **Identifying the Sequence of Absolute Values**:
- The absolute values of the sequence are: \(8, 16, 24, 32, 40, 48, \ldots\).
- These values appear to increase by a consistent amount, i.e., the increments are \(8, 8, 8, 8, 8, \ldots\).
- Therefore, the absolute value can be expressed as a linear function of the index \(n\).
3. **Finding the Linear Function**:
- To find the pattern in the absolute values, we note that:
- At \(n = 0\): \(8 = 8(1)\)
- At \(n = 1\): \(16 = 8(2)\)
- At \(n = 2\): \(24 = 8(3)\)
- At \(n = 3\): \(32 = 8(4)\)
- At \(n = 4\): \(40 = 8(5)\)
- At \(n = 5\): \(48 = 8(6)\)
- We can see that the term can be represented as:
\[
8(n + 1)
\]
4. **Combining the Results**:
- Therefore, the \(n^{th}\) term of the sequence can be expressed as:
\[
a_n = (-1)^n \cdot 8(n + 1)
\]
Thus, the general function for the sequence is:
\[
a_n = (-1)^n \cdot 8(n + 1)
\]