This Fall, The Sports Team Was Trying To Raise Some Money

This fall the sports team were trying to raise some money for their ne

This fall the sports team were trying to raise some money for their new uniforms. They decided to sell Day Lilies. Mr. Rivera purchased a beautiful pink Day Lily and noticed that his garden of day lilies was very crowded. He remembered that five years ago, he started with the one Day Lily that he purchased from school. That fall, Mr. Rivera had three plants because the original plant had two new "baby" plants. In fact, every year each plant had two babies.

How many plants did he have in his garden this fall at the end of the fifth year? How many plants will he have at the end of 10 years?

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The growth of Mr. Rivera's Day Lily garden can be modeled as a geometric progression, where each existing plant produces two new plants every year. Given that he started with a single plant five years ago, we can analyze the number of plants over time using exponential growth principles.

Initially, at year zero, Mr. Rivera had one plant. However, the problem states that in the first fall, the garden contained three plants. This indicates that the initial plant (year zero) produced two new plants, leading to a total of three plants at the end of the first year. From this point onward, the pattern continues: each plant produces two new plants every year, and these new plants are added to the garden.

To formalize this, we recognize that the total number of plants each year can be expressed as follows: The number of new plants produced each year equals twice the number of existing plants in the previous year. Therefore, this process results in exponential growth, and the total number of plants after n years can be modeled as:

N_n = N₀ * 2ⁿ

Where N₀ is the initial number of plants at year zero, which was 1, and n is the number of years after that initial point. Since at the end of the first year (after the initial planting), we have three plants, the initial condition aligns with N₁ = 3, which suggests that the growth begins with N₀ = 1, and the first year’s growth results in three plants.

Alternatively, we can consider the recursive process: Each year, the total number of plants is the existing number plus twice the existing number, which simplifies to:

N_n = N_n-1 + 2 N_n-1 = 3 N_n-1

The initial condition indicates that at the start (year zero), N₀ = 1, and after one year, N₁ = 3 * N₀ = 3. Continuing this pattern, the total number of plants at the end of n years is:

N_n = N₀ * 3ⁿ

Given N₀ = 1, the formula becomes:

N_n = 3ⁿ

Now, to find the number of plants at the end of the fifth year:

N₅ = 3⁵ = 243

And at the end of 10 years:

N₁₀ = 3¹⁰ = 59,049

In conclusion, Mr. Rivera will have 243 plants in his garden at the end of the fifth year, and 59,049 plants at the end of the tenth year, assuming the pattern continues consistently with each plant producing two new plants annually.

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