What can you say about a sequence if you know that each term is a weighted average of the terms to either side?
For example, in the sequence 1, 2, 4, 8, 16, … each term is exactly 2/3rds of the previous term, plus 1/3rd of the following term. What other sequences have exactly that property?
For a given value p, what sequences s1, s2, s3, … sn have the property that each sk = p s(k-1) + (1-p) s(k+1) ? Does knowing s1 also fix the remaining terms?
Amusingly, this actually will help with last weeks puzzle!