There are N periods in the economy. Suppose that we have expense need of E₀, E₁,..., E_N, where E_n is your expense on date = n. Let r be interest rate and Sn, n=0,...,N be the amount you deposit into your saving account on date = n. Initially, there is no money in the account. You finance the spending by withdrawing the same amount from the saving account. Therefore, 

 

the account balance at the end of date 0 = S₀ - E^₀

the account balance at the end of date 1 = (S₀ - E₀)(1+r)+ S₁ - E₁ , ..., and so forth. 

 

In order to find a saving schedule, S₀, S_1,..., S_N, that is just enough for financing all of your expense need E₀, E₁,..., E_N, you want to set the account balance at the end of date N (ending balance) equal to zero. [If the balance is negative, you don't have enough money to pay all of your expenses; You are broke. If it is positive, you have saved too much and gotten extra money.] 

 

Show that "the ending balance =0" is equivalent to "PV of saving = PV of expense."