SOLUTION: your sequence is as follows 1;5;14;30;55 and 8^th term is 204. a)what type of sequence is this? b)find the nth formula

Question 1110479: your sequence is as follows 1;5;14;30;55 and 8^th term is 204.
a)what type of sequence is this?
b)find the nth formula
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
1, 5, 14, 30, 55
:
This sequence is solved by using "Level of Differences" approach
:
Level 1 differences are 4, 9, 16, 25
Level 2 differences are 5, 7, 9
Level 3 differences are 2, 2
:
The Level 3 differences are the same, this tells us that the degree of the nth solution formula is 3
;
a) the nth solution formula is a cubic equation
:
b) the general form for a cubic solution formula for the nth term is
An^3 + Bn^2 + Cn +D = nth term of sequence, where A, B, C, D are real numbers
:
We use the first 4 terms in the sequence
:
A +B +C +D = 1
8A +4B +2C +D = 5
27A +9B +3C +D = 14
64A +16B +4C +D = 30
:
Now use your favorite solver for 4 linear equations in 4 unknowns
:
A = 1/3, B = 1/2, C = 1/6, D = 0
:
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x(n) = n^3/3 +n^2/2 +n/6
:
we are given the 8th term is 204, so check with our formula
:
x(8) = (8^3/3) +(8^2/2) + 8/6
x(8) = (512/3) +(64/2) +(8/6)
x(8) = (1024/6) +(192/6) +(8/6)
x(8) = 1224/6 = 204
:
the formula checks
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