To represent the number of circles at Stage 10 based on the given numbers (10, 1024, 2024, 3024), we can observe a pattern in the numbers provided.
If we analyze the differences between consecutive stages:
- From 10 to 1024, the difference is \(1024 - 10 = 1014\).
- From 1024 to 2024, the difference is \(2024 - 1024 = 1000\).
- From 2024 to 3024, the difference is \(3024 - 2024 = 1000\).
Based on this, it seems that after the first stage (Stage 1), the growth stabilizes to an increment of 1000 for the subsequent stages.
If we denote the number of circles at Stage \(n\) as \(C(n)\), we can express it as follows:
- \(C(1) = 10\)
- \(C(2) = C(1) + 1014 = 1024\)
- \(C(3) = C(2) + 1000 = 2024\)
- \(C(4) = C(3) + 1000 = 3024\)
This suggests a general formula. Given the pattern, we can derive a formula for Stage 10:
\[ C(n) = 10 + 1014 + 1000 \times (n - 2) \quad \text{for } n \geq 2 \]
For \(n = 10\): \[ C(10) = 10 + 1014 + 1000 \times (10 - 2) \] \[ C(10) = 10 + 1014 + 8000 \] \[ C(10) = 10 + 1014 + 8000 = 9024 \]
Thus, the expression for the number of circles at Stage 10 is:
\[ C(10) = 9024 \]
1 answer