To construct an approximate polynomial expression that represents a sequence with constant 4th differences of 48, follow these steps:
Step 1: Write down the given sequence
Write down the given sequence as a table or a list, noting the terms and their corresponding positions (indices). Let's assume the sequence is as follows:
Term: 1 2 3 4 5 6 7 ...
Value: a b c d e f g ...
Step 2: Calculate the 1st differences
Determine the 1st differences by subtracting consecutive terms in the sequence. Add a new row or column to the table representing the 1st differences. For example:
Term: 1 2 3 4 5 6 7 ...
Value: a b c d e f g ...
1st diff: b-a c-b d-c e-d f-e g-f ...
Step 3: Calculate the 2nd differences
Determine the 2nd differences by subtracting consecutive terms in the 1st differences column. Add a new row or column to the table for the 2nd differences. For example:
Term: 1 2 3 4 5 6 7 ...
Value: a b c d e f g ...
1st diff: b-a c-b d-c e-d f-e g-f ...
2nd diff: c-2b+a d-2c+b e-2d+c f-2e+d g-2f+e ...
Step 4: Calculate the 3rd differences
Determine the 3rd differences by subtracting consecutive terms in the 2nd differences column. Add a new row or column to the table for the 3rd differences. For example:
Term: 1 2 3 4 5 6 7 ...
Value: a b c d e f g ...
1st diff: b-a c-b d-c e-d f-e g-f ...
2nd diff: c-2b+a d-2c+b e-2d+c f-2e+d g-2f+e ...
3rd diff: d-3c+3b-a e-3d+3c-b f-3e+3d-c g-3f+3e-d ...
Step 5: Calculate the 4th differences
Determine the 4th differences by subtracting consecutive terms in the 3rd differences column. Add a new row or column to the table for the 4th differences. For example:
Term: 1 2 3 4 5 6 7 ...
Value: a b c d e f g ...
1st diff: b-a c-b d-c e-d f-e g-f ...
2nd diff: c-2b+a d-2c+b e-2d+c f-2e+d g-2f+e ...
3rd diff: d-3c+3b-a e-3d+3c-b f-3e+3d-c g-3f+3e-d ...
4th diff: e-4d+6c-4b+a f-4e+6d-4c+b g-4f+6e-4d+c ...
Step 6: Write the general expression for the 4th differences
Since the 4th differences are constant at 48, the general expression for the 4th differences would be:
4th diff = 48
Step 7: Construct the polynomial expression
Using the general expression for the 4th differences, construct a polynomial expression that represents the sequence. This can be done by repeatedly integrating the 4th differences until reaching the original sequence.
For example, integrating the 4th differences once will yield the 3rd differences:
e-4d+6c-4b+a = 48
To integrate, add a new row or column for the integrated terms and perform cumulative sums:
Term: 1 2 3 4 5 6 7 ...
Value: a b c d e f g ...
1st diff: b-a c-b d-c e-d f-e g-f ...
2nd diff: c-2b+a d-2c+b e-2d+c f-2e+d g-2f+e ...
3rd diff: d-3c+3b-a e-3d+3c-b f-3e+3d-c g-3f+3e-d ...
4th diff: e-4d+6c-4b+a f-4e+6d-4c+b g-4f+6e-4d+c ...
5th diff: f-5e+10d-10c+5b-a g-5f+10e-10d+5c-b ...
By integrating again, the 5th differences can be obtained. This process can be continued until the original sequence is reached.
Note: The specific form of the polynomial expression will depend on the initial values of the sequence (e.g., a, b, c, d, e, f, g, etc.).
Explain the steps to construct an approximate polynomial expression to represent a sequence with constant 4th differences of 48
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