where a 1 (with a - 1 as the interest rate of investment) and where u(k) is the amount spent in year

where α 1 (with α − 1 as
the interest rate of investment) and where u(k) is the amount spent in year k.
He decides that the enjoyment achieved in year k can be expressed as
ψ(u(k)) where ψ, his utility function, is a smooth function, and that
his total lifetime enjoyment is

(a) Find the general optimality relationship for this .

(b) Find the solution for the special case ψ(u) = u1/2.