The powers of a function f: A → A are defined recursively by(a) [BB] Let an be the number...

The powers of a function f: A → A are defined recursively by

(a) [BB] Let an be the number of ways of forming a line of n people distinguished only by sex. For example, there are four possible lines of two people, MM, MF, FM, FF. so a2 = 4. Find a recurrence relation satisfied by an and identify the sequence a1, a2, a3, …

(b) Let an be the number of ways in which a line of n people can be formed such that no two males are standing beside each other. For example, a3 = 5 because there are five ways to form lines of three people with no two males beside each other: FFF, MFF, FMF, FFM, MFM. Find a recurrence relation satisfied by an and identify the sequence a1, a2, a3, ….