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].
