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?
+26400 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?
\(\small{ \begin{array}{llll} \text{after } 1 \text{ day }:& 0.05 \\ \text{after } 2 \text{ days }: & 0.10 &=& 0.05 \cdot 2 \\ \text{after } 3 \text{ days }: & 0.20 = 0.10\cdot 2 = 0.05 \cdot 2 \cdot 2 &=& 0.05 \cdot 2^2\\ \text{after } 4 \text{ days }: & 0.40 = 0.20\cdot 2 =0.05 \cdot 2^2 \cdot 2 \cdot 2 &=& 0.05 \cdot 2^3\\ \dots \\ \text{after } 21 \text{ days }:&&& 0.05 \cdot 2^{21-1}\\ &&& = 0.05 \cdot 2^{20} = 0.05\cdot 1048576 = 52428.80\\ \end{array} }\)
He earn after 21 days 52428.80
+26400 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
\(a_1 = 0.05\)
\(r = 2\)
\(\begin{array}{rcll} s &=& a_1 \left( \frac{1-r^{21}}{1-r} \right) \\ s &=& a_1 \left( \frac{r^{21}-1}{r-1} \right) \\ s &=& 0.05 \left( \frac{2^{21}-1}{2-1} \right) \\ s &=& 0.05 \left( \frac{2^{21}-1}{1} \right) \\ s &=& 0.05 \cdot 2^{21} \\ s &=& 0.05 \cdot 2097152 \\ s &=& 104857.60 \end{array}\)
+26400 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?
without mistake:
all days together:
geometric sequence
\(a_1 = 0.05\)
\(r = 2\)
\(\begin{array}{rcll} s &=& a_1 \left( \frac{1-r^{21}}{1-r} \right) \\ s &=& a_1 \left( \frac{r^{21}-1}{r-1} \right) \\ s &=& 0.05 \left( \frac{2^{21}-1}{2-1} \right) \\ s &=& 0.05 \left( \frac{2^{21}-1}{1} \right) \\ s &=& 0.05 \cdot ( 2^{21} - 1) \\ s &=& 0.05 \cdot 2097151 \\ s &=& 104857.55 \end{array}\)
