Intertemporal Consumption Smoothing

An investor plans to manage an initial wealth of initial wealth alongside an anticipated annual income of income for a total of periods years. The investor applies a discount factor of discount rate and has investments that yield an interest rate of interest rate. Assume the initial wealth and the first income are available at the start of the first year, each income arrives and each year's consumption takes place at the start of that year, savings earn the interest rate between years, and nothing is left at the end. Lifetime utility is the discounted sum over the years of u(c) = c^(1-g)/(1-g), where g is the coefficient of relative risk aversion and each later year is weighted by one more power of the discount factor. Each option lists consumption in each year, in order. Given the investor’s risk aversion coefficient is gamma, how should the investor calculate the optimal smoothed consumption for each year to optimize their discounted lifetime satisfaction?