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A major fishing company does its fishing in a local lake. The first year of the company's operations it managed to catch 170,000 fish. Due to population decreases, the number of fish the company was able to catch decreased by 8% each year. How many total fish did the company catch over the first 16 years, to the nearest whole number? Answer:

A major fishing company does its fishing in a local lake. The first year of the company's operations it managed to catch 170170,000000 fish. Due to population decreases, the number of fish the company was able to catch decreased by 8% 8 \% each year. How many total fish did the company catch over the first 1616 years, to the nearest whole number?\newlineAnswer:

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Q. A major fishing company does its fishing in a local lake. The first year of the company's operations it managed to catch 170170,000000 fish. Due to population decreases, the number of fish the company was able to catch decreased by 8% 8 \% each year. How many total fish did the company catch over the first 1616 years, to the nearest whole number?\newlineAnswer:
  1. Identify initial amount and rate: Identify the initial amount of fish caught and the annual decrease rate.\newlineThe initial amount of fish caught is 170,000170,000, and the annual decrease rate is 8%8\%.
  2. Determine formula for calculation: Determine the formula to calculate the number of fish caught each year. The number of fish caught each year follows a geometric sequence where each term is 92%92\% (100%−8%100\% - 8\%) of the previous term. The formula for the nnth term of a geometric sequence is an=a1⋅r(n−1)a_n = a_1 \cdot r^{(n-1)}, where a1a_1 is the first term, rr is the common ratio, and nn is the term number.
  3. Calculate total over 1616 years: Calculate the total number of fish caught over 1616 years.\newlineWe need to sum up the number of fish caught each year for 1616 years. This is the sum of a finite geometric series.\newlineThe sum of the first nn terms of a geometric series is given by Sn=a1⋅(1−rn)/(1−r)S_n = a_1 \cdot (1 - r^n) / (1 - r), where SnS_n is the sum of the first nn terms.
  4. Substitute values into formula: Substitute the values into the sum formula.\newlinea1=170,000a_1 = 170,000, r=0.92r = 0.92, and n=16n = 16.\newlineS16=170,000×(1−0.9216)/(1−0.92)S_{16} = 170,000 \times (1 - 0.92^{16}) / (1 - 0.92)
  5. Calculate sum S16S_{16}: Calculate the sum S16S_{16}.S16=170,000×(1−0.9216)/(1−0.92)S_{16} = 170,000 \times (1 - 0.92^{16}) / (1 - 0.92)S16=170,000×(1−0.285203)/0.08S_{16} = 170,000 \times (1 - 0.285203) / 0.08S16=170,000×(0.714797)/0.08S_{16} = 170,000 \times (0.714797) / 0.08S16=170,000×8.9349625S_{16} = 170,000 \times 8.9349625S16≈1,518,943.625S_{16} \approx 1,518,943.625
  6. Round total to nearest whole: Round the total number of fish to the nearest whole number.\newlineThe total number of fish caught over the first 1616 years, rounded to the nearest whole number, is approximately 1,518,9441,518,944.

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