3071417125 Assuming That the Patterna Continues the Eighth Term Should Be
Use differences to find a pattern in the sequence.
3,11, 18 , 33 , 65 , 123 , 216
Assuming that the pattern continues, the eighth term should be =
Best Answer
+124657 3 11 18 33 65 123 216
Frst differences 8 7 15 32 58 93
Second differences -1 8 17 26 35
Third differences 9 9 9 9
Since we have a constant difference in the third differences, this series can be represented by a cubic polynomial =
an^2 + bn^2 + cn + d where "n" is is the nth value of the sequence
We have that
a + b + c + d = 3 → -a - b - c - d = -3 (1)
8a + 4b + 2c + d = 11 (2)
27a + 9b + 3c + d = 18 (3)
64a + 16b + 4c + d = 33 (4)
Add (1) and (2), (1) and (3) and (1) and (4)....and we have
7a + 3b + c = 8 (5)
26a + 8b + 2c = 15 (6)
63a + 15b + 3c = 30 (7)
Multiply (5) by -2 add it to (6) and (5) by -3 and add it to (7)
And we have
12a + 2b = - 1 (8)
42a + 6b = 6 (9)
Multiply (8) by - 3 and add it to (9)
6a = 9 → a = 9/6 = 3/2
And to find b we have that 12 (3/2) + 2b = - 1 → 2b = -19 → b = -19/2
And to find c we have 7(3/2) + 3(-19/2) + c = 8 → -36/2 + c = 8 →
-18 + c = 8 → c = 26
And to find d we have that
(3/2) + (-19/2) + 26 + d = 3
-16/2 + 26 + d = 3
-8 + 26 + d = 3
18 + d = 3
c = -15
So....our generating function is (3/2)n^3 - (19/2)n^2 + 26n - 15
3,11, 18 , 33 , 65 , 123 , 216
8, 7, 15, 32, 58, 93
-1, 8, 17, 26, 35
9, 9, 9, 9.........., so, your sequence would continue like this:
3, 11, 18, 33, 65, 123, 216, 353, 543, 795........etc.
P.S. You could also use this formula to give any term that you want:
a(n) = 1/2 (3n^3 - 19n^2 + 52n - 30)
+124657 3 11 18 33 65 123 216
Frst differences 8 7 15 32 58 93
Second differences -1 8 17 26 35
Third differences 9 9 9 9
Since we have a constant difference in the third differences, this series can be represented by a cubic polynomial =
an^2 + bn^2 + cn + d where "n" is is the nth value of the sequence
We have that
a + b + c + d = 3 → -a - b - c - d = -3 (1)
8a + 4b + 2c + d = 11 (2)
27a + 9b + 3c + d = 18 (3)
64a + 16b + 4c + d = 33 (4)
Add (1) and (2), (1) and (3) and (1) and (4)....and we have
7a + 3b + c = 8 (5)
26a + 8b + 2c = 15 (6)
63a + 15b + 3c = 30 (7)
Multiply (5) by -2 add it to (6) and (5) by -3 and add it to (7)
And we have
12a + 2b = - 1 (8)
42a + 6b = 6 (9)
Multiply (8) by - 3 and add it to (9)
6a = 9 → a = 9/6 = 3/2
And to find b we have that 12 (3/2) + 2b = - 1 → 2b = -19 → b = -19/2
And to find c we have 7(3/2) + 3(-19/2) + c = 8 → -36/2 + c = 8 →
-18 + c = 8 → c = 26
And to find d we have that
(3/2) + (-19/2) + 26 + d = 3
-16/2 + 26 + d = 3
-8 + 26 + d = 3
18 + d = 3
c = -15
So....our generating function is (3/2)n^3 - (19/2)n^2 + 26n - 15
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