To find an accurate formula for the given sequence \(2, 3.5, 6.125, 10.71875\), we will analyze the pattern in the numbers.
- Look for patterns:
- The first term is \(2\).
- The second term is \(3.5 = 2 \times 1.75\).
- The third term is \(6.125 = 3.5 \times 1.75\).
- The fourth term is \(10.71875 = 6.125 \times 1.75\).
It appears that each term is obtained by multiplying the previous term by a constant factor \(1.75\).
- Identify the pattern in terms of a formula:
- The first term \(a_1 = 2\).
- The second term \(a_2 = a_1 \times 1.75 = 2 \times 1.75\).
- The third term \(a_3 = a_2 \times 1.75 = (2 \times 1.75) \times 1.75 = 2 \times (1.75^2)\).
- The fourth term \(a_4 = a_3 \times 1.75 = (2 \times (1.75^2)) \times 1.75 = 2 \times (1.75^3)\).
Thus, we can generalize the formula:
\[ a_n = 2 \times (1.75)^{(n-1)} \]
where \(n\) is the term number.
This formula effectively describes the sequence.