An author published a book which was being sold online. The first month the author sold 25300 books, but the sales were declining steadily at \( 10 \% \) each month. If this trend continues, how many total books would the author have sold over the first 20 months, to the nearest whole number?

To calculate the total number of books sold over the first 20 months with a 10% decline each month, we can model the sales using a geometric series. The first month the author sold 25,300 books, and each subsequent month, sales decrease by 10%. This means in the second month, the sales would be: Sales for month 2 = 25,300 * (1 - 0.10) = 25,300 * 0.90 = 22,770 books. Continuing this, the series of sales for the subsequent months will be: - Month 1: 25,300 - Month 2: 22,770 - Month 3: 20,493 - Month 4: 18,443.7 - ... - Month 20: 25,300 * (0.9)^(19) The total sales after 20 months can be calculated using the formula for the sum of a geometric series: \[ S_n = a \frac{1 - r^n}{1 - r} \] where: - \( S_n \) is the sum of the series, - \( a \) is the first term (25,300), - \( r \) is the common ratio (0.90), - \( n \) is the number of terms (20). Plugging in the values: \[ S_{20} = 25300 \frac{1 - (0.90)^{20}}{1 - 0.90} \] Calculating \( (0.90)^{20} \approx 0.1216 \): \[ S_{20} = 25300 \frac{1 - 0.1216}{0.10} \] \[ S_{20} = 25300 \frac{0.8784}{0.10} \] \[ S_{20} = 25300 \times 8.784 \] \[ S_{20} \approx 222,828.92 \] Rounding to the nearest whole number gives approximately 222,829 books sold over the first 20 months.