Geometric series word problems: hike | Algebra 2 | Khan Academy

We're told Sloan went on a four-day hiking trip. Each day she walked 20 more than the distance that she walked the day before. She walked a total of 27 kilometers. What is the distance Sloan walked on the first day of the trip? It says to round our final answer to the nearest kilometer. So, like always, have a go at this and see if you can figure out how much she walked on the first day.

All right, well, let's just call the amount that she walked on the first day "a." Now, using "a," let's see if we can set up an expression for how much she walked in total, and then that should be equal to 27. Hopefully, we're going to be able to solve for "a."

So, the first day she walks "a" kilometers. Now how about the second day? Well, they tell us that each day she walked 20 percent more than the distance she walked the day before. So, on the next day, she's going to walk 20 percent more than "a" kilometers. That's 1.2 times "a."

And what about the day after that, her third day? Well, that's just going to be 1.2 times the second day, and so that's going to be 1.2 times 1.2, or we could say 1.2 squared times "a."

And then how much on the fourth day? She went on a four-day hiking trip, so that's the last day. Well, that's going to be 1.2 times the third day, so that's going to be 1.2 to the third power times "a."

So this is an expression in "a" on how much she walked over the four days, and we know that she walked a total of 27 kilometers. So this is going to be equal to 27 kilometers.

Now, you could solve for "a" over here. You could factor out the "a," and you could say "a times (1 + 1.2 + 1.2 squared + 1.2 to the third power) is equal to 27." Then you could say that "a is equal to 27 over (1 + 1.2 + 1.2 squared + 1.2 to the third power)." We would need a calculator to evaluate this, but I'm going to do a different technique—a technique that would work even if you had 20 terms here.

In theory, you could also do this with 20 terms, but it gets a lot more complicated, or if you had 200 terms. So the other way to approach this is to use the formula for a finite geometric series. What does it evaluate to? Just as a reminder, the sum of the first n terms is going to be the first term, which we could call "a," minus the first term times our common ratio.

In this case, our common ratio is 1.2 because every successive term is 1.2 times the first. So, our first term times our common ratio to the n-th power, all of that over 1 minus the common ratio. In other videos, we explain where this comes from; we prove this. But here, we can just apply it.

We already know what our "a" is. I use that as our variable. Our common ratio in this situation is going to be equal to 1.2, and our "n" is going to be equal to 4. Another way I like to think about it is it's our first term, which we see right over there, minus the term that we did not get to. If we were to have a fifth term, it would have been that fifth term that we're subtracting.

Because we aren't getting to a fourth power here, the fifth term would have been to the fourth power—all of that over one minus the common ratio. So this left-hand side of our equation, we could rewrite as our first term minus our first term times our common ratio (1.2 to the fourth power), all of that over 1 minus our common ratio.

Then, that could be equal to 27. Let me scroll down a little bit so we have some more space to then solve this.

And so let's see, I can simplify this a little bit. This is going to be equal to negative 0.2. Our numerator we can factor out an "a," and so this is going to be equal to "a times (1 - 1.2 to the fourth power)."

Let's see, we can multiply both the numerator and the denominator by a negative one, and so this would get us to "a times (1.2 to the fourth power - 1) over 0.2" is equal to 27.

Again, all I did is take the "a" out of the fraction so it's out here, and I multiplied the numerator and the denominator by a negative. The numerator multiplied by negative would swap these two. Multiplying negative 0.2 times negative is just positive 0.2.

Now, I can just multiply both sides by the reciprocal of this. So, I'll do it here: "0.2 over (1.2 to the fourth - 1)." That cancels with that; that cancels with that. That’s exactly why I did that, and we're left with "a is equal to"—I'll just write it in yellow—"27 times 0.2 all of that over (1.2 to the fourth - 1)."

This expression should give us the exact same value as that expression we just saw, but this is useful even if we had 100 terms; we could use this.

So, I'll get the calculator out. This will give us—so I will actually evaluate this denominator first. So, I'll have "1.2 to the fourth power, which is equal to minus 1," is equal to—now it's in the denominator, so I could just take the reciprocal of it and then multiply that times "27 times 0.2," is equal to 5.029.

Now, they want us to round our answer to the nearest kilometer, so this is going to be approximately equal to five kilometers. That's how much approximately that she would have traveled on the first day of her hiking trip.