To determine who is correct regarding the 2nd differences for the sequence defined by the formula \( a_n = n^2 - 3 \), we need to compute the first differences and then the second differences.
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Compute the values of \( a_n \) for the first few values of \( n \):
- 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 \( n = 4 \): \[ a_4 = 4^2 - 3 = 16 - 3 = 13 \]
So we have the following sequence: \[ a_1 = -2, \quad a_2 = 1, \quad a_3 = 6, \quad a_4 = 13 \]
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Compute the first differences (\( \Delta a_n = a_{n+1} - a_n \)):
- \( \Delta a_1 = a_2 - a_1 = 1 - (-2) = 3 \)
- \( \Delta a_2 = a_3 - a_2 = 6 - 1 = 5 \)
- \( \Delta a_3 = a_4 - a_3 = 13 - 6 = 7 \)
The first differences are: \[ 3, \quad 5, \quad 7 \]
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Compute the second differences (\( \Delta^2 a_n = \Delta a_{n+1} - \Delta a_n \)):
- \( \Delta^2 a_1 = \Delta a_2 - \Delta a_1 = 5 - 3 = 2 \)
- \( \Delta^2 a_2 = \Delta a_3 - \Delta a_2 = 7 - 5 = 2 \)
The second differences are: \[ 2, \quad 2 \]
Since the second differences are constant and equal to \( 2 \), Anjali is correct in stating that the second differences are a constant value of 2.