Two Monthly Savings Plans: Arithmetic versus Geometric Growth
A market trader in Buea joins a monthly savings scheme. Under Plan A she deposits 25,000 FCFA in month 1 and increases the deposit by 2,500 FCFA every month. A microfinance officer offers Plan B: start with 20,000 FCFA in month 1 and increase each later deposit by 8% compared with the previous month. Both plans run for 24 months, and no interest is added beyond the growing deposits. The trader wants to know which plan gives the larger total after 24 months. She also wants to know when the monthly deposit under Plan B first exceeds the monthly deposit under Plan A, because she may switch plans at that point. The officer warns that a larger final total does not necessarily mean Plan B is better in every month. The trader asks for a clear month-by-month comparison, rounded where necessary, and an explanation of the sequence type used in each plan.
1. Calculate the total saved under Plan A after 24 months.
Read the worked answer
Plan A is an arithmetic sequence with first term a = 25,000 FCFA, common difference d = 2,500 FCFA, and n = 24 terms. The sum is S_24 = n/2 [2a + (n − 1)d] = 24/2 [2(25,000) + 23(2,500)] = 12[50,000 + 57,500] = 12 × 107,500 = 1,290,000 FCFA.
So Plan A saves 1,290,000 FCFA over 24 months.
2. Calculate the total under Plan B after 24 months and compare it with Plan A.
Read the worked answer
Plan B is geometric with first term a = 20,000 FCFA and common ratio r = 1.08. The total after 24 months is S_24 = a(r^24 − 1)/(r − 1) = 20,000(1.08^24 − 1)/0.08.
Now 1.08^24 ≈ 6.3412, so S_24 ≈ 20,000(5.3412)/0.08 = 20,000 × 66.765 ≈ 1,335,300 FCFA.
Plan B gives about 1,335,300 FCFA, which is larger than Plan A by about 1,335,300 − 1,290,000 = 45,300 FCFA. However, Plan B starts lower and only overtakes Plan A later, so the final total alone does not show which plan is better in every month.
3. Find the first month in which Plan B's monthly deposit exceeds Plan A's monthly deposit.
Read the worked answer
The monthly deposits are A_n = 25,000 + 2,500(n − 1) B_n = 20,000(1.08)^(n − 1).
Test around the crossover. For n = 15: A_15 = 25,000 + 2,500(14) = 60,000 FCFA. B_15 = 20,000(1.08)^14 ≈ 20,000 × 2.9372 = 58,744 FCFA, which is still below 60,000.
For n = 16: A_16 = 25,000 + 2,500(15) = 62,500 FCFA. B_16 = 20,000(1.08)^15 ≈ 20,000 × 3.1722 = 63,444 FCFA, which is above 62,500.
Since the geometric deposit is growing proportionally and the arithmetic deposit is growing by a fixed amount, once Plan B overtakes Plan A it stays ahead. Therefore the first month is month 16.