To determine which of the differences between consecutive terms is constant, let's analyze the sequences for \( y \) based on the provided values:
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The values of \( y \) correspond to the squares of \( x \):
- \( y(0) = 0^2 = 0 \)
- \( y(1) = 1^2 = 1 \)
- \( y(2) = 2^2 = 4 \)
- \( y(3) = 3^2 = 9 \)
- \( y(4) = 4^2 = 16 \)
- \( y(5) = 5^2 = 25 \)
- \( y(6) = 6^2 = 36 \)
- \( y(7) = 7^2 = 49 \)
- \( y(8) = 8^2 = 64 \)
- \( y(9) = 9^2 = 81 \)
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1st differences: The 1st differences are obtained by subtracting each term from the next: \[ \begin{align*} y(1) - y(0) &= 1 - 0 = 1 \ y(2) - y(1) &= 4 - 1 = 3 \ y(3) - y(2) &= 9 - 4 = 5 \ y(4) - y(3) &= 16 - 9 = 7 \ y(5) - y(4) &= 25 - 16 = 9 \ y(6) - y(5) &= 36 - 25 = 11 \ y(7) - y(6) &= 49 - 36 = 13 \ y(8) - y(7) &= 64 - 49 = 15 \ y(9) - y(8) &= 81 - 64 = 17 \ \end{align*} \] The 1st differences are: \( 1, 3, 5, 7, 9, 11, 13, 15, 17 \). These differ and are not constant.
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2nd differences: The 2nd differences are obtained by subtracting the 1st difference from the next: \[ \begin{align*} 3 - 1 &= 2 \ 5 - 3 &= 2 \ 7 - 5 &= 2 \ 9 - 7 &= 2 \ 11 - 9 &= 2 \ 13 - 11 &= 2 \ 15 - 13 &= 2 \ 17 - 15 &= 2 \ \end{align*} \] The 2nd differences are: \( 2, 2, 2, 2, 2, 2, 2, 2 \). These are constant.
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3rd differences: The 3rd differences are obtained by subtracting the 2nd difference from the next: \[ 2 - 2 = 0 \quad \text{(and all other pairs are also 0)} \] The 3rd differences are: \( 0, 0, 0, 0, 0, 0, 0 \). These are also constant.
Given this analysis, both the 2nd difference and 3rd difference are constant. However, when looking at the options:
- The 1st difference is not constant.
- The 2nd difference is constant.
- The 3rd difference is constant.
So, the correct answer to which differences between consecutive terms is constant is:
Option #2: 2nd difference.