Let's address each part of the problem:
i) Find the number of students after 2 years.
The growth formula is given by \( \text{Future Value} = \text{Present Value} \times (1 + \text{Growth Rate})^{\text{Number of Years}} \).
For this problem, the present value (initial number of students) is 1000, the growth rate is 20% or 0.20, and the number of years is 2.
\[ \text{Future Value} = 1000 \times (1 + 0.20)^2 \]
\[ \text{Future Value} = 1000 \times (1.20)^2 \]
\[ \text{Future Value} = 1000 \times 1.44 \]
\[ \text{Future Value} = 1440 \]
So, after 2 years, the number of students will be 1440.
ii) In which year, either the second or third, more students are added?
The growth is always in terms of the initial number of students. If you calculate the number of students after the second year, you can compare it with the number of students after the third year. We already found that after 2 years, there are 1440 students. Now, let's find the number of students after 3 years:
\[ \text{Future Value after 3 years} = 1000 \times (1 + 0.20)^3 \]
\[ \text{Future Value after 3 years} = 1000 \times (1.20)^3 \]
\[ \text{Future Value after 3 years} = 1000 \times 1.728 \]
\[ \text{Future Value after 3 years} = 1728 \]
So, in the third year, more students are added (1728 compared to 1440 in the second year).
iii) If the number of students is increased by 15% in the second year and 20% in the first year, in which year are more students added?
Let's calculate the number of students in the second year with an additional increase of 15%:
\[ \text{Future Value after 2 years (with 15% increase)} = 1000 \times (1 + 0.20)^2 \times (1 + 0.15) \]
\[ \text{Future Value after 2 years (with 15% increase)} = 1440 \times 1.15 \]
\[ \text{Future Value after 2 years (with 15% increase)} = 1656 \]
Now, let's calculate the number of students in the first year with an additional increase of 20%:
\[ \text{Future Value after 1 year (with 20% increase)} = 1000 \times (1 + 0.20) \times (1 + 0.20) \]
\[ \text{Future Value after 1 year (with 20% increase)} = 1000 \times 1.20 \times 1.20 \]
\[ \text{Future Value after 1 year (with 20% increase)} = 1440 \]
So, more students are added in the second year with a 15% increase compared to the first year with a 20% increase.
