USE DIFFERENCES TO FOUND A PATTERN IN THE SQUENCE
5,14,29,59,113,200, 329
ASSUMING THAT THE PATTERN EIGHTH TERM SHOULD BE
5 14 29 59 113 200 329
9 15 30 54 87 129
6 15 24 33 42
9 9 9 9
We have three rows of non-zero differences
So.....we can create a 3rd degree polynomial ⇒ an^3 + bn^2 + cn + d
(Where n is the nth term )
We have this system
a + b + c + d = 5
8a + 4b + 2c + d = 14
27a + 9b + 3c + d = 29
64a + 16b + 4c + d = 59
This is a little tedious to solve ( but not impossible), so I'll enlist a little help with technology here :
https://matrix.reshish.com/gauss-jordanElimination.php
a = 3/2 b = -6 c = 33/2 d = -7
So.....the polynomial that will generate this series is
P(n) = (3/2)n^3 -6n^2 + (33/2)n - 7
The 8th term = 509