A worker is paid 0.05 on the first day, 0.10 on the second day, 0.20 on the third day and 0.40 on the fourth day, and so on. How much money in total does the work earn after working 21 days?
\(\begin{array}{llll} \text{after } 1 \text{ day }:& 0.05 \\ \text{after } 2 \text{ days }: & 0.10 &=& 0.05 \cdot 2 \\ \text{after } 3 \text{ days }: & 0.20 = 0.10\cdot 2 = 0.05 \cdot 2 \cdot 2 &=& 0.05 \cdot 2^2\\ \text{after } 4 \text{ days }: & 0.40 = 0.20\cdot 2 =0.05 \cdot 2^2 \cdot 2 \cdot 2 &=& 0.05 \cdot 2^3\\ \dots \\ \text{after } 21 \text{ days }:&&& 0.05 \cdot 2^{21-1} = 0.05 \cdot 2^{20} = 0.05\cdot 1048576 = 52428.80\\ \end{array}\)
all days together:
geometric sequence
\(a_1 = 0.05\)
\(r = 2\)
\(\begin{array}{rcll} s &=& a_1 \left( \frac{1-r^{21}}{1-r} \right) \\ s &=& a_1 \left( \frac{r^{21}-1}{r-1} \right) \\ s &=& 0.05 \left( \frac{2^{21}-1}{2-1} \right) \\ s &=& 0.05 \left( \frac{2^{21}-1}{1} \right) \\ s &=& 0.05 \cdot 2^{21} \\ s &=& 0.05 \cdot 2097152 \\ s &=& 104857.60 \end{array}\)
