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Sagot :
To determine the next number in the sequence [tex]\(0, 14, 76, 246, 606, 1260, 2334\)[/tex] using the method of successive differences, we will follow these steps:
1. Calculate the first order differences (Δ1): Subtract each term from the next term.
[tex]\[ 14 - 0 = 14 \\ 76 - 14 = 62 \\ 246 - 76 = 170 \\ 606 - 246 = 360 \\ 1260 - 606 = 654 \\ 2334 - 1260 = 1074 \][/tex]
So, the first order differences are:
[tex]\[ [14, 62, 170, 360, 654, 1074] \][/tex]
2. Calculate the second order differences (Δ2): Subtract each term in the first order differences from the next term.
[tex]\[ 62 - 14 = 48 \\ 170 - 62 = 108 \\ 360 - 170 = 190 \\ 654 - 360 = 294 \\ 1074 - 654 = 420 \][/tex]
So, the second order differences are:
[tex]\[ [48, 108, 190, 294, 420] \][/tex]
3. Calculate the third order differences (Δ3): Subtract each term in the second order differences from the next term.
[tex]\[ 108 - 48 = 60 \\ 190 - 108 = 82 \\ 294 - 190 = 104 \\ 420 - 294 = 126 \][/tex]
So, the third order differences are:
[tex]\[ [60, 82, 104, 126] \][/tex]
4. Calculate the fourth order differences (Δ4): Subtract each term in the third order differences from the next term.
[tex]\[ 82 - 60 = 22 \\ 104 - 82 = 22 \\ 126 - 104 = 22 \][/tex]
So, the fourth order differences are:
[tex]\[ [22, 22, 22] \][/tex]
Since all the fourth order differences are constant (22), the differences appear to be consistent.
5. Determine the next fourth order difference: Given that the fourth order differences are constant, the next fourth order difference is also [tex]\(22\)[/tex].
6. Calculate the next third order difference: Add the next fourth order difference to the last third order difference.
[tex]\[ 126 + 22 = 148 \][/tex]
7. Calculate the next second order difference: Add the next third order difference to the last second order difference.
[tex]\[ 420 + 148 = 568 \][/tex]
8. Calculate the next first order difference: Add the next second order difference to the last first order difference.
[tex]\[ 1074 + 568 = 1642 \][/tex]
9. Calculate the next term in the sequence: Add the next first order difference to the last term in the original sequence.
[tex]\[ 2334 + 1642 = 3976 \][/tex]
So, the next number in the sequence is [tex]\(\boxed{3976}\)[/tex].
1. Calculate the first order differences (Δ1): Subtract each term from the next term.
[tex]\[ 14 - 0 = 14 \\ 76 - 14 = 62 \\ 246 - 76 = 170 \\ 606 - 246 = 360 \\ 1260 - 606 = 654 \\ 2334 - 1260 = 1074 \][/tex]
So, the first order differences are:
[tex]\[ [14, 62, 170, 360, 654, 1074] \][/tex]
2. Calculate the second order differences (Δ2): Subtract each term in the first order differences from the next term.
[tex]\[ 62 - 14 = 48 \\ 170 - 62 = 108 \\ 360 - 170 = 190 \\ 654 - 360 = 294 \\ 1074 - 654 = 420 \][/tex]
So, the second order differences are:
[tex]\[ [48, 108, 190, 294, 420] \][/tex]
3. Calculate the third order differences (Δ3): Subtract each term in the second order differences from the next term.
[tex]\[ 108 - 48 = 60 \\ 190 - 108 = 82 \\ 294 - 190 = 104 \\ 420 - 294 = 126 \][/tex]
So, the third order differences are:
[tex]\[ [60, 82, 104, 126] \][/tex]
4. Calculate the fourth order differences (Δ4): Subtract each term in the third order differences from the next term.
[tex]\[ 82 - 60 = 22 \\ 104 - 82 = 22 \\ 126 - 104 = 22 \][/tex]
So, the fourth order differences are:
[tex]\[ [22, 22, 22] \][/tex]
Since all the fourth order differences are constant (22), the differences appear to be consistent.
5. Determine the next fourth order difference: Given that the fourth order differences are constant, the next fourth order difference is also [tex]\(22\)[/tex].
6. Calculate the next third order difference: Add the next fourth order difference to the last third order difference.
[tex]\[ 126 + 22 = 148 \][/tex]
7. Calculate the next second order difference: Add the next third order difference to the last second order difference.
[tex]\[ 420 + 148 = 568 \][/tex]
8. Calculate the next first order difference: Add the next second order difference to the last first order difference.
[tex]\[ 1074 + 568 = 1642 \][/tex]
9. Calculate the next term in the sequence: Add the next first order difference to the last term in the original sequence.
[tex]\[ 2334 + 1642 = 3976 \][/tex]
So, the next number in the sequence is [tex]\(\boxed{3976}\)[/tex].
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