A Puzzle A Day: 2026-05-10

InnovateCo, a promising tech startup, launched its premium analytics service. For their initial clients, they implemented a unique pricing strategy: the first client paid $100, the second paid $120, the third paid $140, and so on, with each subsequent client paying $20 more than the previous one. After successfully acquiring their 100th client with this model, the company decided to adjust its strategy. All *future* clients (starting from the 101st) would pay a flat rate that is 20% less than the *average price paid by the first 100 clients*. How much will the 101st client pay for the premium analytics service?
Correct: $872
This problem involves an arithmetic progression and percentage calculation. 1. **Identify the arithmetic progression:** The first term (a₁) = $100 The common difference (d) = $20 The number of terms (n) = 100 2. **Calculate the price paid by the 100th client (a_100):** The formula for the nth term of an arithmetic progression is a_n = a₁ + (n-1)d. a_100 = 100 + (100-1) * 20 a_100 = 100 + 99 * 20 a_100 = 100 + 1980 a_100 = $2080 3. **Calculate the total revenue from the first 100 clients (S_100):** The formula for the sum of an arithmetic progression is S_n = n/2 * (a₁ + a_n). S_100 = 100/2 * (100 + 2080) S_100 = 50 * 2180 S_100 = $109,000 4. **Calculate the average price paid by the first 100 clients:** Average Price = Total Revenue / Number of Clients Average Price = 109000 / 100 Average Price = $1090 5. **Calculate the price for the 101st client:** The 101st client pays 20% less than the average price. Price for 101st client = Average Price - (0.20 * Average Price) Price for 101st client = 1090 - (0.20 * 1090) Price for 101st client = 1090 - 218 Price for 101st client = $872 Therefore, the 101st client will pay $872.
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