To analyze the polynomial sequence given by \( a_n = 2n^4 - n^3 \) and determine at which differences it reaches a constant value, we can start by calculating the differences of the sequence.
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First Difference: The first difference is given by \( a_{n+1} - a_n \): \[ a_{n+1} = 2(n+1)^4 - (n+1)^3 \] Expanding each part: \[ (n+1)^4 = n^4 + 4n^3 + 6n^2 + 4n + 1 \] Therefore: \[ 2(n+1)^4 = 2(n^4 + 4n^3 + 6n^2 + 4n + 1) = 2n^4 + 8n^3 + 12n^2 + 8n + 2 \] Next, we expand \( (n+1)^3 \): \[ (n+1)^3 = n^3 + 3n^2 + 3n + 1 \] Therefore: \[
- (n+1)^3 = -n^3 - 3n^2 - 3n - 1 \] Putting it all together: \[ a_{n+1} = 2n^4 + 8n^3 + 12n^2 + 8n + 2 - n^3 - 3n^2 - 3n - 1 \] Simplifying: \[ a_{n+1} = 2n^4 + (8n^3 - n^3) + (12n^2 - 3n^2) + (8n - 3n) + (2 - 1) \] \[ = 2n^4 + 7n^3 + 9n^2 + 5n + 1 \]
Now, the first difference \( a_{n+1} - a_n \) is: \[ a_{n+1} - a_n = (2n^4 + 7n^3 + 9n^2 + 5n + 1) - (2n^4 - n^3) \] \[ = 7n^3 + 9n^2 + 5n + 1 + n^3 = 8n^3 + 9n^2 + 5n + 1 \]
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Second Difference: Next, we compute the second difference \( \Delta^2 a_n = \Delta(a_{n+1} - a_n) = a_{n+2} - a_{n+1} - (a_{n+1} - a_n) \).
To find \( a_{n+2} \): \[ a_{n+2} = 2(n+2)^4 - (n+2)^3 \] Expanding: \[ (n+2)^4 = n^4 + 8n^3 + 24n^2 + 32n + 16 \Rightarrow 2(n+2)^4 = 2n^4 + 16n^3 + 48n^2 + 64n + 32 \] \[ (n+2)^3 = n^3 + 6n^2 + 12n + 8 \Rightarrow - (n+2)^3 = -n^3 - 6n^2 - 12n - 8 \] Combining: \[ a_{n+2} = 2n^4 + 16n^3 + 48n^2 + 64n + 32 - n^3 - 6n^2 - 12n - 8 \] \[ = 2n^4 + (16n^3 - n^3) + (48n^2 - 6n^2) + (64n - 12n) + (32 - 8) \] \[ = 2n^4 + 15n^3 + 42n^2 + 52n + 24 \] Thus, \[ a_{n+2} - a_{n+1} = (2n^4 + 15n^3 + 42n^2 + 52n + 24) - (2n^4 + 7n^3 + 9n^2 + 5n + 1) \] \[ = (15n^3 - 7n^3) + (42n^2 - 9n^2) + (52n - 5n) + (24 - 1) \] \[ = 8n^3 + 33n^2 + 47n + 23 \]
Now the second difference is: \[ \Delta^2 a_n = (8n^3 + 33n^2 + 47n + 23) - (8n^3 + 9n^2 + 5n + 1) \] \[ = (33n^2 - 9n^2) + (47n - 5n) + (23 - 1) = 24n^2 + 42n + 22 \]
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Third Difference: Continuing this process, we compute: \[ \Delta^3 a_n = \Delta(24n^2 + 42n + 22) = (24(n+1)^2 + 42(n+1) + 22) - (24n^2 + 42n + 22) \] We can compute: \[ 24(n^2 + 2n + 1) + 42(n + 1) + 22 - (24n^2 + 42n + 22) \] \[ = (24n^2 + 48n + 24 + 42n + 42 + 22) - (24n^2 + 42n + 22) \] \[ = 48n + 24 + 42 + 22 \] \[ = 24n + 58 \]
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Fourth Difference: Next, we compute: \[ \Delta^4 a_n = \Delta(48n + 58) = (48(n+1) + 58) - (48n + 58) \] \[ = 48n + 48 + 58 - (48n + 58) = 48 \]
Thus, the fourth difference is constant, which indicates that the polynomial sequence \( a_n = 2n^4 - n^3 \) reaches a constant value at the fourth difference.