Auditing a Savings Plan Against a Rising Bill
A household sets up a monthly savings plan. In month 1 they deposit $120, and each month they increase the deposit by $15 over the previous month, so month 2 is $135, month 3 is $150, and so on. At the same time, their electricity bill starts at $95 in month 1 and rises by 6% each month, compounded. The household wants to know the total saved over 12 months, the electricity bill in month 12, and the first month in which the monthly saving deposit exceeds the monthly electricity bill by at least $100. They also want to know whether the plan is realistic if their monthly income is fixed at $3,200 and other essential spending is $2,750 per month.
1. Find the total saved over 12 months and the month-12 deposit.
Read the worked answer
The deposits form an arithmetic sequence with first term a = 120 and common difference d = 15.
Month-12 deposit = 120 + 11 x 15 = 120 + 165 = $285.
Sum of 12 terms = (n/2)(first + last) = (12/2)(120 + 285) = 6 x 405 = $2,430.
Check by the other formula: (12/2)(2 x 120 + 11 x 15) = 6 x (240 + 165) = 6 x 405 = $2,430, consistent.
So the household saves $2,430 over the year, with the final monthly deposit at $285.
2. Compute the electricity bill in month 12 and identify the first month in which the deposit exceeds the bill by at least $100.
Read the worked answer
Bill is geometric: 95 x 1.06^(n-1).
Month 12: 1.06^11. Compute stepwise: 1.06^2=1.1236; ^4=1.2625; ^8=1.5938; ^11 = 1.5938 x 1.1236 x 1.06 = 1.5938 x 1.1910 = 1.8982. Bill = 95 x 1.8982 = $180.33, about $180.
Now test months where deposit - bill >= 100. Deposit(n) = 120 + 15(n-1) = 105 + 15n.
Month 1: 120 - 95 = 25. Month 3: 150 - 95 x 1.1236 = 150 - 106.74 = 43.26. Month 5: 180 - 95 x 1.2625 = 180 - 119.94 = 60.06. Month 7: 210 - 95 x 1.4185 = 210 - 134.76 = 75.24. Month 9: 240 - 95 x 1.5938 = 240 - 151.41 = 88.59. Month 11: 270 - 95 x 1.7908 = 270 - 170.13 = 99.87, just under 100. Month 12: 285 - 180.33 = 104.67, at least 100.
So month 12 is the first month meeting the condition, and only narrowly (month 11 misses by 13 cents).
3. Given fixed income of $3,200 and other essentials of $2,750, assess whether the plan is sustainable through month 12.
Read the worked answer
Discretionary money available each month = 3200 - 2750 = $450, before electricity.
In month 1: deposit 120 plus bill 95 = $215, leaving 450 - 215 = $235 spare. Sustainable.
In month 12: deposit 285 plus bill 180.33 = $465.33, which exceeds the $450 available by $15.33. The plan breaks in month 12.
Find the first month where deposit + bill > 450. Month 10: deposit 255; bill = 95 x 1.06^9 = 95 x 1.6895 = 160.50; total 415.50, fine. Month 11: deposit 270; bill 170.13; total 440.13, fine. Month 12: 465.33, over.
So the plan is affordable for 11 months but not the 12th. The judgement is that the arithmetic progression outpaces the fixed discretionary budget once the compounded bill is added, and the household should cap the monthly increment at about $10, or pause the escalation in month 12.