10. Show that an = : 2n5n is also a solution to the recurrence relation an = 7an-1-10an-2. What would the initial conditions need to be for this to be the closed formula for the sequence?

(a) Male honeybees are from single parent families; they are from unfertilized eggs so only have mothers. Female honeybees are from two parent families; they are from fertilized eggs so have both mothers and fathers. Let Bn bethe number of nth generation ancestors of a male honeybee. For example: B1 = 1 (itself), B2 = 1 (its mother), B3 = 2 (its grand-father and grand-mother), etc. Find a recurrence relation for Bn. [Assume that no honeybee appears more than once in the ancestral tree; that is, each honeybee has at most one child in the tree.](b) Prove that if n is divisible by four then Bn is divisible by three.

Solve the recurrence relation, given: • ao = 3 • a₁ = 4 . • an = 5an-1-6an-2 Remember: • The theorem is an = c₁an-1 + c₂an-2. -b± √b² - 4ac 2a • The quadratic formula is O a O b Oc Od an = 5(4)" -2(3)" an = 5(2)"-2(3)" an = 5(2)"+2(3)" an = 5(3)" -2(3)"

Find the general solution of the recursive relation an an = a10 (3)" + α20 (-3)" an = (a10+ a11 n) (-3)" None of these. an = α10 (6)" + α20 (-9)" an = (α10+ α11 n) (3)" = 6an-19an-2, n ≥ 2.