Today we're going to talk about the sum of a geometric sequence in real world situations. So we're going to do a couple of examples involving real world situations, both with finite geometric sequences and with infinite geometric sequences.
So let's start by looking at a real world situation involving the sum of a finite geometric sequence. So let's suppose I have a bank account and I deposit $1,500 into that account. The account earns 4.2% interest compounded annually, or just one time per year.
And I know that in my account I'm going to be adding an additional deposit at the beginning of every year of $1,500. So we can represent this situation using a geometric sequence. So the first term in my sequence is going to be $1,500. So this term represents the initial deposit made with no interest.
The second term in my sequence is going to be 1,563. So now this term represents the initial amount plus 1 year's worth of interest. And our first term is the additional deposit value that's made after one year.
The third term in our sequence is going to be 1,628.65. And now this value represents our initial deposit plus 2 year's worth of interest. The second term can now be thought of as the previous deposit plus 1 year's worth of interest. And the first term can be thought of as the next deposit, which does not have any interest yet.
So now let's see how we can use our formula for the sum of a finite geometric sequence to find the value of our account after a certain number of years. So let's say we want to find the value of our account after 5 years, and we're going to round that to the nearest whole dollar.
So using my formula, I want to find the sum of the first 5 terms. I want to find the value after 5 years. My first term, a1, is 1,500. My value for r, my common ratio, I can find by taking any term in the sequence and dividing it by the one before it.
So if I take 1563 and divide that by 1,500, that's going to give me 1.042 for my common ratio. So I'll substitute that in for r. And my exponent, n, is going to be 5. And in my denominator, I'll have 1 minus 1.04 again.
I'm going to start by simplifying in my numerator. 1.04 to the 5th power is going to give me approximately 1.2284. So I'll bring down the rest of my terms.
Then I'm going to subtract in my numerator and in my denominator. And that's going to give me in my numerator, I'll have negative 0.2284. And in my denominator, after subtracting, I'll negative 0.042.
Dividing these two numbers, I find that it's equal to about 5.438. And finally, multiplying these two values together, I find that the value of my account rounded to the nearest whole dollar is approximately $8,157 after 5 years.
So now let's do an example, a real world example, involving the sum of the infinite geometric sequence. Suppose there's a rocket that has a top speed of 8 kilometers per second. However, it decelerates at a rate of 20% every second. So we can write a sequence to represent this situation.
So the first term in my sequence is going to be 8. 8 kilometers per second is the speed that we're starting with. If it's decreasing at a rate of 20%, that means the value of the next term is going to be 80% of the term before. It is decreasing by 20%.
So that's going to be 6.4 kilometers per second. The next value also decreases by 20%. So it's going to be 80% of the value before. So that's going to be 5.12. And our fourth term would be 4.096 and this would continue on.
Now this is an infinite geometric sequence. Each term is getting closer and closer to 0, but it's never going to actually equal 0. No term will ever actually equal 0. And because of this, the sum of the infinite sequence is going to converge to a specific value.
So we can use the formula for the sum of the infinite geometric sequence to find this value. So we have our sequence that we just wrote out. We're going to use the formula for the sum of the infinite geometric sequence to determine the value, or the distance, that the rocket has traveled before it's virtually stopped.
So the sum is going to be equal to a1 is going to be 8. And my value for r is going to be 0.8, 80%. So now I'm going to start to simplify this by subtracting in my denominator. That will give me 0.2.
1/0.2 is going to give me 5. And 8 times 5 is going to give me 40. So I found that it'll be 40 kilometers before the rocket has virtually stopped.
So let's go over our key points from today. Geometric sequences can be used to model real world situations such as the value of a bank account and the rate of change of velocity of an object in motion. In the formulas for the sum of a finite geometric sequence, and an infinite geometric sequence, a1 is the first term in the sum, r is the common ratio between consecutive terms, and n is the number of terms.
So I hope that these key points and examples helps you understand a little bit more about geometric sequences in real world situations. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.