Question

Пруд зарастает водяными лилиями. На первый день выросла одна лилия, на второй день — две лилии, на третий день — четыре лилии, на четвёртый день — восемь лилий, ... и так на каждый следующий день число лилий удваивалось. Наконец, на 42 день пруд зарос полностью. На который день пруд зарос наполовину? (запиши номер дня)

Answer 1

На 41 день пруд зарос на половину

Answer 2

Solution:

To solve this problem, we can use the concept of exponential growth. The number of lilies doubles each day, leading to a pattern that can be represented by the formula 2^(n-1), where n is the number of days.

1. Pattern of Lily Growth: - On the first day, there is 1 lily. - On the second day, there are 2 lilies. - On the third day, there are 4 lilies. - On the fourth day, there are 8 lilies. - This pattern continues, with the number of lilies doubling each day.

2. Formula for Lily Growth: - The formula for the number of lilies on the nth day is given by 2^(n-1), where n is the number of days.

3. Finding the Day When the Pond is Half Covered: - We want to find the day when the pond is half covered, which means we need to find the value of n for which 2^(n-1) is half of the total lilies on the 42nd day.

4. Calculating the Day: - Using the formula 2^(n-1), we can calculate the value of n for which the number of lilies is half of the total lilies on the 42nd day.

Calculation:

- The total number of lilies on the 42nd day is 2^(42-1) = 2^41. - To find the day when the pond is half covered, we need to find the value of n for which 2^(n-1) = (1/2) * 2^41.

- Solving for n: - 2^(n-1) = (1/2) * 2^41 - 2^(n-1) = 2^40 - n - 1 = 40 - n = 41

Answer:

The pond is half covered on the 41st day.