To find the explicit formula for the arithmetic sequence \(-11, -3, 5, 13, \ldots\), let's first identify the first term and the common difference.
- The first term \( b(1) \) is \(-11\).
- To find the common difference \(d\), subtract the first term from the second term: \[ d = -3 - (-11) = -3 + 11 = 8 \]
Now we have:
- First term \( b(1) = -11 \)
- Common difference \( d = 8 \)
The formula for the \(n\)-th term of an arithmetic sequence can be expressed as: \[ b(n) = b(1) + (n-1)d \]
Substituting the values we found: \[ b(n) = -11 + (n - 1) \cdot 8 \]
Now simplify this: \[ b(n) = -11 + 8(n - 1) \] \[ b(n) = -11 + 8n - 8 \] \[ b(n) = 8n - 19 \]
Therefore, the explicit formula for the \(n\)-th term of the given arithmetic sequence is: \[ b(n) = 8n - 19 \]