To find a recursive formula for the given sequence \(2, 7, 12, 17, 22, 27\), we need to observe how the sequence progresses.
Let's first identify the pattern of the sequence:
- The first term \(a_1 = 2\).
- The second term \(a_2 = 7\).
- The third term \(a_3 = 12\).
- The fourth term \(a_4 = 17\).
- The fifth term \(a_5 = 22\).
- The sixth term \(a_6 = 27\).
Next, let's look at the differences between successive terms:
- \(a_2 - a_1 = 7 - 2 = 5\)
- \(a_3 - a_2 = 12 - 7 = 5\)
- \(a_4 - a_3 = 17 - 12 = 5\)
- \(a_5 - a_4 = 22 - 17 = 5\)
- \(a_6 - a_5 = 27 - 22 = 5\)
We see that the difference between consecutive terms is consistently \(5\). This indicates that the sequence is an arithmetic sequence with a common difference of \(5\).
Using this information, we can write the recursive formula for the sequence:
\[ a_n = a_{n-1} + 5 \quad \text{for } n \geq 2 \] with the initial condition: \[ a_1 = 2. \]
Thus, the complete recursive formula for the sequence is: \[ \begin{align*} a_1 &= 2, \ a_n &= a_{n-1} + 5 \quad \text{for } n \geq 2. \end{align*} \]