The number of students in a school increases by 10 percent every year, and there are currently 2,988 students. How many students will the school have after six years?
We are told that there are currently 2,988 students in a school. The number of students increases by 10 percent every year. 10 percent of 2,988 is 298.8. Adding this to the initial number of students means there would be 3286.8 students in year two. As we cannot have a fraction of a student, we would need to round this to the nearest whole number.
If a number increases by 10 percent, this is the same as multiplying the initial number by 1.1, as 1.1 is equivalent to 110 percent. In this question, we need to calculate the number of students in the school after six years. One way of doing this would be to simply continue to multiply by 1.1. However, a quicker way would be to recognize that we have a geometric sequence. Any sequence is geometric if it has a common ratio or multiplier between consecutive terms. In this case, we have a common ratio of 1.1.
The first term of our geometric sequence, denoted 𝑎 sub one, is 2,988. And we know that the general or 𝑛th term of a geometric sequence, denoted 𝑎 sub 𝑛, is equal to 𝑎 sub one multiplied by 𝑟 to the power of 𝑛 minus one. We want to calculate 𝑎 sub six, the sixth term of the sequence. This is equal to 2,988 multiplied by 1.1 to the power of six minus one, which is the same as 1.1 to the fifth power. Typing this into our calculator gives us 4,812.203 and so on. As already mentioned, the number of students must be an integer value. Rounding this to the nearest whole number gives us 4,812.
After six years, the school will have 4,812 students.