The population of a city is now 844501 and increases at a constant rate of 12 percent per year. Find the population after eight years giving the answer to the nearest integer.
So, the first thing to look at is the fact that the increase is at a constant rate. So, we know the population is increasing at a constant rate. And that rate is 12 percent per year. Well, with this question, we can think about it in a couple of ways, both of them giving us the same equation and the same answer. The first way is to consider it as a geometric sequence. Well, as it’s in a geometric sequence, and we know that cause it’s increasing at a constant rate, then we have a general form for this. And that is that 𝑎 sub 𝑛, so any term, is equal to 𝑎 sub one, our first term, multiplied by 𝑟, a common ratio, to the power of 𝑛 minus one, where 𝑛 is the term number.
Well, our first term is 844501 because that’s our first value. And our 𝑟 is 1.12. But how do we get that? Well, we’re looking to increase at a constant rate of 12 percent per year. Well, if we’re looking to increase by 12 percent each year, then it’s 100 percent, it’s the amount we had, add 12 percent means 112 percent of the amount that they had at the beginning of that previous year. Well, as percent means out of 100, then 112 percent is equal to 112 over 100, which is equal to 1.12, which is our multiplier. Okay, great!
And we also know that 𝑛 is equal to eight because we’re looking for the population after eight years. Okay, so, now, we can use this to solve the problem. Because what we can do is substitute this into our general form. So, when do, we get the eight term, or the population after eight years, is equal to 844501 multiplied by 1.12 to the power of eight minus one. So therefore, this is gonna be the same as 844501 multiplied by 1.12 to the power of seven. Well, this is gonna give the answer 1866922.65 continued.
But we want this to the nearest integer. So, what we’re gonna do is round. And because the first decimal place is a six, so it’s five or above, we’re gonna round the last digit, so the unit, from a two to a three. So therefore, we found that the population after eight years is 1866923. And that’s to the nearest integer.
Well, I did mention there’s another way to think about this question. And the other way to think about it is compound interest. And as a compound interest question, we have a formula. And that formula is that 𝐴, the amount that we’re looking for, is equal to 𝑃, our initial or principal amount, multiplied by. And then, we’ve got one plus 𝑟, where 𝑟 is the interest rate as a decimal. And then, this is raised to the power of 𝑡, which is the number of time periods.
Well, if we take a look at our problem, our 𝑃, so our initial amount, is 844501. Our 𝑟, which is our interest rate as a decimal, is gonna be 0.12. And that’s because 12 percent means 12 out of 100 or 12 divided by 100. And our 𝑡 is gonna be equal to seven because there are seven time periods up until the eighth year, which is what we’re looking at. And if we think about how that’d work, if we think about the first year we’re in, it’s year one. So then, we could see that we’d have seven periods of interest, cause you get the interest annually, till we get to year eight.
Okay, great. So, we’ve got our values. So, we can put them now into the formula. So, when we do that, we get our population after eight years is gonna be equal to 844501 multiplied by one plus 0.12 to the power of seven. Well, we can see that from this, we arrive at exactly the same formula that we got when we used the geometric sequence method. And that is 844501 multiplied by 1.12 to the power of seven. So therefore, we’re gonna get to the same answer. And we can say that definitely the population after eight years is 1866923 to the nearest integer.