1. **State the problem:** Ludek has two payment options for 8 days of work. Option 1 pays a fixed $40 per day. Option 2 pays doubling amounts starting at $2 on day 1, $4 on day 2, $8 on day 3, etc. We want to find total earnings for each day up to day 8 and compare them. 2. **Formulas:** - Option 1 total earnings after $n$ days: $$E_1(n) = 40 \times n$$ - Option 2 daily payment is geometric: payment on day $k$ is $$2^k$$ dollars starting from $k=1$ with $2^1=2$. Total earnings after $n$ days is sum of geometric series: $$E_2(n) = 2 + 4 + 8 + \cdots + 2^n = \sum_{k=1}^n 2^k$$ 3. **Sum of geometric series:** $$E_2(n) = 2(2^n - 1)$$ This comes from the formula for sum of powers of 2 starting at $2^1$. 4. **Calculate total earnings for each day 1 to 8:** - Option 1: $$E_1(1) = 40, E_1(2) = 80, E_1(3) = 120, E_1(4) = 160, E_1(5) = 200, E_1(6) = 240, E_1(7) = 280, E_1(8) = 320$$ - Option 2: $$E_2(1) = 2(2^1 - 1) = 2(2 - 1) = 2$$ $$E_2(2) = 2(2^2 - 1) = 2(4 - 1) = 6$$ $$E_2(3) = 2(2^3 - 1) = 2(8 - 1) = 14$$ $$E_2(4) = 2(2^4 - 1) = 2(16 - 1) = 30$$ $$E_2(5) = 2(2^5 - 1) = 2(32 - 1) = 62$$ $$E_2(6) = 2(2^6 - 1) = 2(64 - 1) = 126$$ $$E_2(7) = 2(2^7 - 1) = 2(128 - 1) = 254$$ $$E_2(8) = 2(2^8 - 1) = 2(256 - 1) = 510$$ 5. **Interpretation:** Option 1 grows linearly by 40 each day. Option 2 grows exponentially, doubling daily payments and summing them. 6. **Scatter plots:** Plot points $(n, E_1(n))$ and $(n, E_2(n))$ for $n=1$ to $8$ on the same axes. Connect points in each plot to show growth trends. **Final answer:** - Option 1 totals: 40, 80, 120, 160, 200, 240, 280, 320 - Option 2 totals: 2, 6, 14, 30, 62, 126, 254, 510 Option 2 surpasses Option 1 after day 6.