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To determine at which differences the given polynomial sequence [tex]\( a_n = 2n^4 - n^3 \)[/tex] reaches a constant value, let's follow the step-by-step process of computing the differences.
1. Identify the first few terms of the sequence:
Let's calculate [tex]\( a_n \)[/tex] for the first five integers [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
[tex]\[ \begin{aligned} a_1 &= 2 \cdot 1^4 - 1^3 = 2 - 1 = 1, \\ a_2 &= 2 \cdot 2^4 - 2^3 = 2 \cdot 16 - 8 = 32 - 8 = 24, \\ a_3 &= 2 \cdot 3^4 - 3^3 = 2 \cdot 81 - 27 = 162 - 27 = 135, \\ a_4 &= 2 \cdot 4^4 - 4^3 = 2 \cdot 256 - 64 = 512 - 64 = 448, \\ a_5 &= 2 \cdot 5^4 - 5^3 = 2 \cdot 625 - 125 = 1250 - 125 = 1125. \end{aligned} \][/tex]
So, the sequence of terms is [tex]\( \{a_1, a_2, a_3, a_4, a_5\} = \{1, 24, 135, 448, 1125\} \)[/tex].
2. First differences:
Calculate the differences between consecutive terms:
[tex]\[ \begin{aligned} a_2 - a_1 &= 24 - 1 = 23, \\ a_3 - a_2 &= 135 - 24 = 111, \\ a_4 - a_3 &= 448 - 135 = 313, \\ a_5 - a_4 &= 1125 - 448 = 677. \end{aligned} \][/tex]
Thus, the first differences are [tex]\( \{23, 111, 313, 677\} \)[/tex].
3. Second differences:
Calculate the differences between consecutive first differences:
[tex]\[ \begin{aligned} 111 - 23 &= 88, \\ 313 - 111 &= 202, \\ 677 - 313 &= 364. \end{aligned} \][/tex]
Therefore, the second differences are [tex]\( \{88, 202, 364\} \)[/tex].
4. Third differences:
Calculate the differences between consecutive second differences:
[tex]\[ \begin{aligned} 202 - 88 &= 114, \\ 364 - 202 &= 162. \end{aligned} \][/tex]
The third differences are [tex]\( \{114, 162\} \)[/tex].
5. Fourth differences:
Calculate the differences between consecutive third differences:
[tex]\[ 162 - 114 = 48. \][/tex]
The fourth differences are [tex]\( \{48\} \)[/tex].
From the calculations above, we see that the fourth differences are constant.
Thus, the sequence [tex]\( a_n = 2n^4 - n^3 \)[/tex] reaches a constant value at the fourth differences.
Answer: 4th differences
1. Identify the first few terms of the sequence:
Let's calculate [tex]\( a_n \)[/tex] for the first five integers [tex]\( n = 1, 2, 3, 4, 5 \)[/tex]:
[tex]\[ \begin{aligned} a_1 &= 2 \cdot 1^4 - 1^3 = 2 - 1 = 1, \\ a_2 &= 2 \cdot 2^4 - 2^3 = 2 \cdot 16 - 8 = 32 - 8 = 24, \\ a_3 &= 2 \cdot 3^4 - 3^3 = 2 \cdot 81 - 27 = 162 - 27 = 135, \\ a_4 &= 2 \cdot 4^4 - 4^3 = 2 \cdot 256 - 64 = 512 - 64 = 448, \\ a_5 &= 2 \cdot 5^4 - 5^3 = 2 \cdot 625 - 125 = 1250 - 125 = 1125. \end{aligned} \][/tex]
So, the sequence of terms is [tex]\( \{a_1, a_2, a_3, a_4, a_5\} = \{1, 24, 135, 448, 1125\} \)[/tex].
2. First differences:
Calculate the differences between consecutive terms:
[tex]\[ \begin{aligned} a_2 - a_1 &= 24 - 1 = 23, \\ a_3 - a_2 &= 135 - 24 = 111, \\ a_4 - a_3 &= 448 - 135 = 313, \\ a_5 - a_4 &= 1125 - 448 = 677. \end{aligned} \][/tex]
Thus, the first differences are [tex]\( \{23, 111, 313, 677\} \)[/tex].
3. Second differences:
Calculate the differences between consecutive first differences:
[tex]\[ \begin{aligned} 111 - 23 &= 88, \\ 313 - 111 &= 202, \\ 677 - 313 &= 364. \end{aligned} \][/tex]
Therefore, the second differences are [tex]\( \{88, 202, 364\} \)[/tex].
4. Third differences:
Calculate the differences between consecutive second differences:
[tex]\[ \begin{aligned} 202 - 88 &= 114, \\ 364 - 202 &= 162. \end{aligned} \][/tex]
The third differences are [tex]\( \{114, 162\} \)[/tex].
5. Fourth differences:
Calculate the differences between consecutive third differences:
[tex]\[ 162 - 114 = 48. \][/tex]
The fourth differences are [tex]\( \{48\} \)[/tex].
From the calculations above, we see that the fourth differences are constant.
Thus, the sequence [tex]\( a_n = 2n^4 - n^3 \)[/tex] reaches a constant value at the fourth differences.
Answer: 4th differences
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