To find the 2nd differences of the sequence given by the formula \( a_n = n^2 - 3 \), we can follow these steps:
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Calculate the first few terms of the sequence using the formula \( a_n = n^2 - 3 \):
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For \( n = 1 \): \[ a_1 = 1^2 - 3 = 1 - 3 = -2 \]
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For \( n = 2 \): \[ a_2 = 2^2 - 3 = 4 - 3 = 1 \]
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For \( n = 3 \): \[ a_3 = 3^2 - 3 = 9 - 3 = 6 \]
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For \( n = 4 \): \[ a_4 = 4^2 - 3 = 16 - 3 = 13 \]
Thus, the first terms of the sequence are: \[ a_1 = -2, \quad a_2 = 1, \quad a_3 = 6, \quad a_4 = 13 \]
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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 \( a_4 - a_3 = 13 - 6 = 7 \)
The first differences are: \[ 3, \quad 5, \quad 7 \]
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Calculate the second differences:
- Second difference \( 5 - 3 = 2 \)
- Second difference \( 7 - 5 = 2 \)
The second differences are constant and equal to: \[ 2, \quad 2 \]
From these calculations, we find that the second differences are indeed a constant value of 2.
Therefore, Anjali is correct in finding the 2nd differences.