The Recurring Deposit That Never Settles
A student is analysing a savings scheme. A deposit of 1000 dollars is made at the start of each year, and the account earns 5 percent interest compounded annually, credited at the end of each year. Let u_n be the balance immediately after the n-th deposit and before that year's interest is credited. The student writes u_1 = 1000 and claims u_{n+1} = 1.05 u_n + 1000. A tutor says the recurrence is wrong because interest is credited after the deposit, so the new deposit should not earn interest in the same step. The student also wants a closed form for u_n and asks whether the balance converges as n grows. The tutor insists the sequence diverges but wants the growth rate quantified.
1. Critique the recurrence u_{n+1} = 1.05 u_n + 1000 given that u_n is measured immediately after the n-th deposit and before interest. State the correct recurrence and explain the timing.
Read the worked answer
Timing analysis. u_n is the balance right after deposit n, before interest for year n. During year n, interest is credited on the whole balance u_n, giving 1.05 u_n at the end of year n. Then deposit n+1 is added at the start of year n+1, so
u_{n+1} = 1.05 u_n + 1000.
The student's recurrence is actually correct under this definition. The tutor's objection would apply only if u_n were defined after interest was credited. If instead v_n denotes the balance just after interest is credited in year n, then v_n = 1.05 u_n and u_{n+1} = v_n + 1000, which is the same relation. The lesson: a recurrence is only meaningful relative to a precise definition of the state variable. The tutor's criticism is a definitional mismatch, not an algebraic error.
2. Solve the recurrence to obtain a closed form for u_n, and determine the long-run behaviour, quantifying the growth.
Read the worked answer
Closed form. The recurrence u_{n+1} = 1.05 u_n + 1000 is linear with fixed point L satisfying L = 1.05L + 1000, so L = -1000/0.05 = -20000. Write u_n = -20000 + c(1.05)^{n-1}. Using u_1 = 1000: 1000 = -20000 + c, so c = 21000. Hence
u_n = 21000(1.05)^{n-1} - 20000.
Check: u_1 = 21000 - 20000 = 1000. u_2 = 21000(1.05) - 20000 = 22050 - 20000 = 2050, matching 1.05(1000) + 1000.
Long-run behaviour. Since 1.05 > 1, (1.05)^{n-1} grows without bound, so u_n -> +infinity. The sequence diverges. The dominant term is 21000(1.05)^{n-1}, so the balance grows geometrically at 5 percent per year asymptotically; the constant -20000 becomes negligible. The tutor is right that it diverges, but the useful statement is the geometric rate, not merely 'diverges'.