To derive the recursive formula from the given formula \( h(n) = -10 + 12n \), we first need to determine the value of \( h(1) \) and how to express \( h(n) \) in terms of \( h(n-1) \).
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Finding \( h(1) \): \[ h(1) = -10 + 12 \cdot 1 = -10 + 12 = 2 \]
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Finding \( h(n) \) in terms of \( h(n - 1) \): First, let's calculate \( h(n-1) \): \[ h(n-1) = -10 + 12(n-1) = -10 + 12n - 12 = -22 + 12n \]
Now we want to find the difference \( h(n) - h(n-1) \): \[ h(n) - h(n-1) = (-10 + 12n) - (-22 + 12n) = -10 + 12n + 22 - 12n = 12 \]
Now that we have this difference, we can express \( h(n) \) in terms of \( h(n - 1) \): \[ h(n) = h(n-1) + 12 \]
So, the complete recursive formula is:
- \( h(1) = 2 \)
- \( h(n) = h(n - 1) + 12 \) for \( n \geq 2 \)