Saving for a Trip: Monthly Plan versus Lump Sum
Niamh is 16 and wants to have money for a trip in two years. She can save €40 in month 1 and increase the amount by €8 each month for 24 months, with no interest. Alternatively, her grandmother offers a lump sum of €3,000 now, which Niamh can place in a two-year deposit paying 2.5% per year compounded annually. Niamh's friend says the monthly plan is better because 'you end up saving more than €3,000', while her grandmother says the lump sum is better because 'interest adds money'. Niamh also wants to know how much she would need to save in month 1 if she wants the 24 monthly deposits to total exactly €3,600 while still increasing by €8 each month. Compare the two options and advise her, noting one important reason the comparison is not completely fair.
1. Calculate the total amount Niamh saves under the 24-month plan.
Read the worked answer
The monthly deposits form an arithmetic sequence with first term a = 40 and common difference d = 8. There are n = 24 terms. The sum is S₂₄ = (24/2)[2(40) + (24 − 1)(8)] = 12[80 + 184] = 12 × 264 = 3168. So Niamh saves €3,168 under the monthly plan, assuming no interest is added.
2. Calculate the value of the €3,000 lump sum after two years at 2.5% per year compounded annually. Compare the two options and advise Niamh.
Read the worked answer
Use F = P(1 + i)^t with P = 3000, i = 0.025 and t = 2. F = 3000(1.025)² = 3000 × 1.050625 = €3,151.875, or about €3,151.88. The monthly plan gives €3,168, which is €16.12 more than the lump sum. Under the stated assumptions, the monthly plan produces the larger total, so the friend's conclusion is correct for these figures, though the grandmother is right that interest adds money. The comparison is not completely fair because the monthly deposits are made over the two years and earn no interest in this plan, while the full €3,000 lump sum earns interest for the entire two years. If the monthly savings also earned interest, the result could change; the monthly plan also requires regular discipline.
3. Find the first monthly deposit needed for the 24 deposits to total exactly €3,600, still increasing by €8 each month.
Read the worked answer
Let the first deposit be a. The sum of 24 terms is S₂₄ = (24/2)[2a + 23(8)] = 12[2a + 184]. Set this equal to 3600: 12[2a + 184] = 3600. Divide by 12: 2a + 184 = 300. Then 2a = 116, so a = 58. Niamh would need to save €58 in month 1, then €66 in month 2, and so on, increasing by €8 each month. Check: 12[2(58) + 184] = 12[116 + 184] = 12 × 300 = 3600.