2,4,8,3,5,14,4,6,22,7,9,...This infinite sequence is a recurrent sequence. The recursive formula is an=−99305an−1−1145an−2+9973an−3+99401an−4+33221an−5+1168, where n>5, a1=2, a2=4, a3=8, a4=3, and a5=5.
RECURSIVE FORMULA
Linear Recurrence Relation
[View Steps]
Given a sequence of 11 terms, the recursive formula is of the form an=x0+x1an−1+...+xkan−k,where 1≤k≤5
When k=5: an=x0+an−1x1+an−2x2+an−3x3+an−4x4+an−5x5,n>5 14462279=x0+5x1+3x2+8x3+4x4+2x5=x0+14x1+5x2+3x3+8x4+4x5=x0+4x1+14x2+5x3+3x4+8x5=x0+6x1+4x2+14x3+5x4+3x5=x0+22x1+6x2+4x3+14x4+5x5=x0+7x1+22x2+6x3+4x4+14x5{x0:1168, x1:−99305, x2:−1145, x3:9973, x4:99401, x5:33221}
The nth term rule is:an=−99305an−1−1145an−2+9973an−3+99401an−4+33221an−5+1168
Given a sequence of m terms, the recursive formula is of the form an=x0+x1an−1+...+xkan−k,where 1≤k≤⌊2m−1⌋
Using all the given terms, solve the systems of equations for xi when k=1,...,⌊2m−1⌋. If xi is not found for any k, a recursive formula cannot be found using this method.
When k=1: an=x0+x1an−1,n>1 Solve for x0 and x1: a2a3⋮am=x0+x1a1=x0+x1a2=x0+x1am−1
