To find the nth term of the sequence \(-7, -2, 3, 8\), let's first analyze the pattern.
- Identify the differences between consecutive terms:
- From \(-7\) to \(-2\): difference = \( -2 - (-7) = 5\)
- From \(-2\) to \(3\): difference = \(3 - (-2) = 5\)
- From \(3\) to \(8\): difference = \(8 - 3 = 5\)
All the differences are constant and equal to \(5\), which indicates that this sequence is an arithmetic sequence with a common difference of \(5\).
- The formula for the nth term of an arithmetic sequence is:
\[ a_n = a_1 + (n - 1)d \]
Where:
- \(a_1\) is the first term of the sequence.
- \(d\) is the common difference.
- \(n\) is the term number.
- For this sequence:
- \(a_1 = -7\)
- \(d = 5\)
Plugging in these values:
\[ a_n = -7 + (n - 1) \cdot 5 \]
- Simplifying this expression:
\[ a_n = -7 + 5n - 5 \] \[ a_n = 5n - 12 \]
So the nth term of the sequence is:
\[ \boxed{5n - 12} \]