To find out who is correct regarding the second differences of the sequence given by the formula \( a_n = n^2 - 3 \), we need to start by calculating the first differences and then the second differences of the sequence.
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Calculate the 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 \( n = 5 \): \( a_5 = 5^2 - 3 = 25 - 3 = 22 \)
Thus, the sequence is: \[ -2, 1, 6, 13, 22 \]
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Calculate the first differences: \[ \text{1st differences}: \]
- \( 1 - (-2) = 3 \)
- \( 6 - 1 = 5 \)
- \( 13 - 6 = 7 \)
- \( 22 - 13 = 9 \)
So, the first differences are: \[ 3, 5, 7, 9 \]
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Calculate the second differences: \[ \text{2nd differences}: \]
- \( 5 - 3 = 2 \)
- \( 7 - 5 = 2 \)
- \( 9 - 7 = 2 \)
Thus, the second differences are: \[ 2, 2, 2 \]
Since all the second differences are the constant value of 2, Anjali is correct.
The final answer is: c) Anjali is correct. Jamal and Shayna both calculated 1st differences.