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An author published a book which was being sold online. The first month the author sold 8000 books, but the sales were declining steadily at 9% each month. If this trend continues, how many total books would the author have sold over the first 21 months, to the nearest whole number? Answer:

An author published a book which was being sold online. The first month the author sold 80008000 books, but the sales were declining steadily at 9% 9 \% each month. If this trend continues, how many total books would the author have sold over the first 2121 months, to the nearest whole number?\newlineAnswer:

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Q. An author published a book which was being sold online. The first month the author sold 80008000 books, but the sales were declining steadily at 9% 9 \% each month. If this trend continues, how many total books would the author have sold over the first 2121 months, to the nearest whole number?\newlineAnswer:
  1. Identify Initial Number: Identify the initial number of books sold and the monthly decline rate.\newlineThe initial number of books sold is 80008000, and the monthly decline rate is 9%9\%.
  2. Determine Formula: Determine the formula for the total number of books sold over a period of time with a steady decline.\newlineThe total number of books sold can be calculated using the formula for the sum of a geometric series: Sn=a(1−rn)1−rS_n = \frac{a(1 - r^n)}{1 - r}, where aa is the first term, rr is the common ratio (1−decline rate)(1 - \text{decline rate}), and nn is the number of terms.
  3. Calculate Common Ratio: Calculate the common ratio for the geometric series.\newlineThe decline rate is 9%9\%, so the common ratio rr is 1−0.09=0.911 - 0.09 = 0.91.
  4. Apply Geometric Series Formula: Apply the formula for the sum of a geometric series to find the total number of books sold over 2121 months.\newlineUsing the formula Sn=a(1−rn)(1−r)S_n = \frac{a(1 - r^n)}{(1 - r)}, where a=8000a = 8000, r=0.91r = 0.91, and n=21n = 21, we get:\newlineS21=8000(1−0.9121)(1−0.91)S_{21} = \frac{8000(1 - 0.91^{21})}{(1 - 0.91)}.
  5. Calculate Sum: Calculate the sum S21S_{21}. \newlineS21=8000(1−0.9121)/(1−0.91)S_{21} = 8000(1 - 0.91^{21}) / (1 - 0.91)\newline=8000(1−0.9121)/0.09= 8000(1 - 0.91^{21}) / 0.09\newlineNow, calculate 0.91210.91^{21} using a calculator.\newline0.9121≈0.14260.91^{21} \approx 0.1426 (rounded to four decimal places)\newlineNow, substitute this value into the formula.\newlineS21=8000(1−0.1426)/0.09S_{21} = 8000(1 - 0.1426) / 0.09\newline=8000(0.8574)/0.09= 8000(0.8574) / 0.09\newline=6859.2/0.09= 6859.2 / 0.09\newline=76213.333…= 76213.333\ldots
  6. Round Total Number: Round the total number of books sold to the nearest whole number.\newlineThe total number of books sold over 2121 months, rounded to the nearest whole number, is approximately 7621376213.

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