To find the second difference of the sequence given by the formula \( a_n = n^2 - 3 \), we first need to determine the values of \( a_n \) for various integers \( n \) and then calculate the first and second differences.
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Calculate the first few terms of the sequence:
- For \( n = 1 \): \[ a_1 = 1^2 - 3 = 1 - 3 = -2 \]
- For \( n = 2 \): \[ a_2 = 2^2 - 3 = 4 - 3 = 1 \]
- For \( n = 3 \): \[ a_3 = 3^2 - 3 = 9 - 3 = 6 \]
- For \( n = 4 \): \[ a_4 = 4^2 - 3 = 16 - 3 = 13 \]
- For \( n = 5 \): \[ a_5 = 5^2 - 3 = 25 - 3 = 22 \]
So, the sequence is: \(-2, 1, 6, 13, 22\).
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Calculate the first differences:
- First difference \( a_2 - a_1 \): \[ 1 - (-2) = 3 \]
- First difference \( a_3 - a_2 \): \[ 6 - 1 = 5 \]
- First difference \( a_4 - a_3 \): \[ 13 - 6 = 7 \]
- First difference \( a_5 - a_4 \): \[ 22 - 13 = 9 \]
The first differences are: \(3, 5, 7, 9\).
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Calculate the second differences:
- Second difference \( (5 - 3) \): \[ 2 \]
- Second difference \( (7 - 5) \): \[ 2 \]
- Second difference \( (9 - 7) \): \[ 2 \]
The second differences are: \(2, 2, 2\).
Since the second difference is constant and equal to \(2\), we find that Angela Lee is correct in her assertion.
Conclusion: Angela Lee is correct; the second difference is a constant value of \(2\).