You open an account at a bank that pays 5 interest yearly an

You open an account at a bank that pays 5% interest yearly, and deposit a₀ dollars in it. Every year you withdraw $10 times the number of years you have had the account. For example, if you started with $1000, then in the first year you would earn $50 in interest and withdraw $10, leaving $1040, and in the second year would earn $52 and withdraw $20, leaving $1072, and so forth.

(a) Find a recurrence for a_n, the balance in the account after n years.

(b) Solve the recurrence to find a closed form for a_n.

(c) What is the smallest initial deposit which would guarantee that the account never runs out of money?

Solution