Answer:
220
Explanation:
You want to use differences to find the pattern and the next term in the sequence 3, 8, 15, 28, 51, 88, 143.
First differences
Subtracting each term from the next, we find the sequence of first differences to be ...
5, 7, 13, 23, 37, 55
Second differences
The first differences are not constant, so we know the sequence is not linear. They are not in an arithmetic progression, so we know the sequence is not quadratic. They do not have a common ratio, so we know the sequence is not exponential.
The second differences are the differences of the sequence of first differences. They are ...
2, 6, 10, 14, 18
Third differences
The differences of the terms of the 2nd-difference sequence are constant: 4.
4, 4, 4, 4, 4
Constant 3rd differences mean the original sequence can be described by a 3rd degree polynomial. The coefficients found by a calculator are shown in the attachment.
Next term
Since we know the third differences are constant, we can work our way back up the chain of differences to find the next term of the original sequence.
next 2nd difference = 18 +4 = 22
next 1st difference = 55 +22 = 77
next sequence term = 143 +77 = 220
The eighth term should be 220.
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Additional comment
The n-th term is ...
f(n) = 2/3n^3 -3n^2 +28/3n -4