To determine who is correct about the 2nd differences for the sequence given by the formula \( a_n = n^2 - 3 \), we can first generate a few terms of the sequence and then calculate the first and second differences.
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Calculate the sequence values: \[ \begin{align*} a_1 &= 1^2 - 3 = 1 - 3 = -2 \ a_2 &= 2^2 - 3 = 4 - 3 = 1 \ a_3 &= 3^2 - 3 = 9 - 3 = 6 \ a_4 &= 4^2 - 3 = 16 - 3 = 13 \ a_5 &= 5^2 - 3 = 25 - 3 = 22 \ \end{align*} \]
The sequence terms are: \(-2, 1, 6, 13, 22\).
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Calculate the 1st differences: \[ \begin{align*} \Delta_1 a_1 &= a_2 - a_1 = 1 - (-2) = 3 \ \Delta_1 a_2 &= a_3 - a_2 = 6 - 1 = 5 \ \Delta_1 a_3 &= a_4 - a_3 = 13 - 6 = 7 \ \Delta_1 a_4 &= a_5 - a_4 = 22 - 13 = 9 \ \end{align*} \]
The 1st differences are: \(3, 5, 7, 9\).
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Calculate the 2nd differences: \[ \begin{align*} \Delta_2 a_1 &= \Delta_1 a_2 - \Delta_1 a_1 = 5 - 3 = 2 \ \Delta_2 a_2 &= \Delta_1 a_3 - \Delta_1 a_2 = 7 - 5 = 2 \ \Delta_2 a_3 &= \Delta_1 a_4 - \Delta_1 a_3 = 9 - 7 = 2 \ \end{align*} \]
The 2nd differences are: \(2, 2, 2\).
This all leads us to conclude that the 2nd differences are indeed constant and equal to 2.
Therefore, Anjali is correct. The correct response is:
Anjali is correct. Jamal and Shayna both calculated 1st differences.