Victoria is training in the gym. On the treadmill, she runs 250 meters in the first minute. And the distance she runs decreases by 10 percent each subsequent minute. How far does she run in 10 minutes? Give your answer to the nearest meter.
Let’s think about what’s happening here. In the first minute, Victoria runs 250 meters. In the second minute, Victoria decreases the number of meters she ran by 10 percent. So we take the meters that she ran in the first minute and we calculate 10 percent of that, 10 percent written as a decimal, as 0.10. 250 times 0.10 is 25. Victoria ran 25 less meters in the second minute. But that means we need to take 250 and subtract 25 which gives us 225 meters. Victoria ran 225 meters in the second minute. Calculating how far she ran in the second minute in this way takes two steps. First you have to find 10 percent and then you have to subtract.
There’s actually a simpler way to do this, so that we only have to do one step. A decrease of 10 percent means that Victoria maintains 90 percent of her speed. And that means we can actually take the 250 meters she ran in the first minute and multiply that by the 90 percent that she maintains. Then we write that as a decimal as 0.90 and 250 times 0.90 is 225 meters. It’s a one-step way of calculating this. Okay, so back to our distances. For minute three, Victoria ran 90 percent of what she ran in minute two. Remember, 90 percent of what she ran in minute two will represent a decrease of 10 percent. 90 percent of 225 is equal to 202.5.
Now, we could continue this process all the way to 10. We would need to do 10 different sets of calculations and then add them all together. But I wonder if we can recognise another pattern here. Remember that 225 was 90 percent of 250. Let’s plug in 90 percent of 250 in the places that we see 225. If minute two was 250 times 0.90, we can rewrite minute three to be 250 times 0.90 times 0.90, 250 times 0.90 squared. In this way, minute two is 250 times 0.90 to the first power. And we could even say that the first minute is 250 times 0.90 to the zero power. Anything to the zero power is one. So 250 times 0.90 to the zero power is equal to 250.
And now we want to look at the connection between the exponent and the minute. In the second minute, we have an exponent of one. In the third minute, we have an exponent of two. What would we expect the equation to calculate the distance that Victoria ran in the fourth minute to be? In the fourth minute, Victoria ran 250 times 0.90 cubed meters. And this type of pattern should make us think of a geometric sequence. In a geometric sequence, 𝑎 sub 𝑛, a term is equal to the first term, 𝑎 sub one times 𝑟, the rate that it’s changing to the 𝑛 minus one power, where 𝑛 is greater than or equal to one.
Now, how does this help us? Well, this fact in and of itself doesn’t necessarily help. But we know a formula for calculating the sum of 𝑛 terms of a geometric sequence. The sum of 𝑛 number of terms is equal to 𝑎 sub one times one minus 𝑟 to the 𝑛 power all over one minus 𝑟, as long as the rate is not one. Let’s try using this formula. We’re looking for the sum of the first 10 values 𝑟𝑎 sub one would be how far Victoria had gone after one minute, which is 250. And then we have one minus the rate. And that’s going to be the 90 percent, the rate that it’s changing, to the 10th power all over one minus 0.90.
If we plug that into our calculator, we get 1628.3039. If we want to round to the nearest meter, we’ll need to look to the digit to the right of that value. In this case, it’s a three. Since that is less than five, we’ll round down, meaning the eight in the ones place stays the same. And we say that Victoria ran 1628 meters in 10 minutes. Now in this question, did you have to recognise that it was a geometric sequence? Well, no. And if you didn’t, here’s what you would have to do.
Let’s go back to our table method. You would have in the first minute she ran 250 meters. And the second minute you’d multiply 0.90 times 250 to get 225. If you’re using a calculator, you can use the answer function so that for the third minute you’ll say it’s 0.90 times the answer. That’s going to be calculating 0.90 times the previous answer of 225, which gives you 202.5. And again, you can continue this process in the calculator by just multiplying 0.90 by the answer, which is the previous option. Once you’ve calculated all 10 of these values, you would add them together. And they would sum to 1628.3039 which we would then round to 1628 meters.