Eight-Year Membership: Geometric Series and Binomial Bound
A sports club advertises two payment plans for an eight-year membership. Plan A charges €600 in the first year, and each later year's fee is 5% higher than the previous year's fee. Plan B charges a fixed €700 each year for eight years. The advertisement says: 'Plan A costs no more than €5,800 in total; use the approximation (1.05)^8≈1.47.' A treasurer checks the exact total before advising members. She also wants a simple error estimate for the advertisement's binomial approximation, because the omitted terms are positive. Members who leave early pay only for the years completed, so the number of years matters as well as the total. The treasurer must decide which plan is cheaper for a member who stays all eight years, and whether the advertised bound is safe for budgeting.
1. Calculate the exact total cost of Plan A for eight years and compare it with Plan B.
Read the worked answer
Plan A is a geometric series with first term 600 and ratio 1.05. The total after 8 years is S₈=600(1.05^8−1)/(1.05−1)=600(1.05^8−1)/0.05. Now 1.05^8=1.4774554437890625, so S₈=600(0.4774554437890625)/0.05=600×9.54910887578125=5729.46532546875. Thus Plan A costs €5,729.47 to the nearest cent. Plan B costs 8×700=€5,600. Plan A is more expensive by €129.47. The advertisement's bound of €5,800 is true for the exact total, but it is not tight.
2. Expand (1.05)^8 using the first three terms of the binomial theorem and explain whether the resulting total is an overestimate or underestimate of Plan A.
Read the worked answer
With x=0.05, (1+x)^8=1+8x+C(8,2)x²+C(8,3)x³+… . Using three terms gives 1+8(0.05)+28(0.05)²=1+0.4+0.07=1.47. All remaining terms are positive because x=0.05>0, so the true value is greater than 1.47. The corresponding total 600(1.47−1)/0.05=600×9.4=€5,640 is therefore a lower estimate, not an upper estimate. The exact total €5,729.47 exceeds it by about €89.47. Using 1.47 as though it were exact would under-budget Plan A; the advertised €5,800 is still a valid upper bound, but it does not follow from that truncated expansion as an upper bound.
3. Compare the plans for a member who stays 7 years and for one who stays 8 years, and state the first integer year for which Plan B is cheaper.
Read the worked answer
For 7 years, Plan A is 600(1.05^7−1)/0.05. Since 1.05^7=1.40710042265625, S₇=600(0.40710042265625)/0.05=600×8.142008453125=€4,885.21. Plan B for 7 years is €4,900, so Plan A is cheaper by €14.79. For 8 years, Plan A is €5,729.47 while Plan B is €5,600, so Plan B is cheaper by €129.47. Therefore the first integer year for which Plan B is cheaper is year 8. The comparison is not simply 'Plan B is always cheaper'; the answer depends on the number of years completed.