Hi and welcome. My name is Anthony Varela. And today, we're going to find the sum of geometric sequences from real world situations. So our first example we're going to go through is a sum of a finite geometric sequence. It's going to be within a financial example. And then our second example is going to be the sum of an infinite geometric sequence. And our example will be the distance of an object.
So let's start with our financial example first. So we put $500 into an investment account. And every year, this gains 3% interest. And it's compounded just one time per year. And at the start of each year, we also put in an additional $500. So I'm going to create a sequence to represent this situation. And we'll be explaining the meaning of each term as we build the sequence.
So the first term is $500. That represents the initial $500 that we've put into the account. Well, after one year, this gains 3% interest. So if I multiply $500 by 1.03, so that represents 100% of the $500 plus an additional 3%, I'm going to have $515.
So that's the second term in the sequence. And that represents our initial $500 having gained interest for one year. And the first term now, what does that represent? That represents the fresh $500 that we just put in at the beginning of this new year.
So another year goes by. So my $515 has gained another 3%. So I have $530.45. This represents my $500 having gained 3% interest for two years. And now, my second term $515, that represents the $500 that has only gained interest for one year. And my first term represents the brand new $500 deposit that I make at the beginning of the year.
So this pattern continues as the years go by. So we have a couple of other terms leading up to year 7 of this type of investment. And now, after seven years, I'd like to know how much money is in this account. Well, that means I'm going to be adding the terms to this sequence.
So to find the sum of this geometric sequence, I'm going to use this equation here where we have the sum of a certain number of terms equals the value of the first term multiplied by 1 over r raised to the n over 1 minus r. Now, r is the common ratio, how we get from one term to the next. And n is the number of terms in our sequence.
So we'd like to find the sum of these seven terms. So here n equals 7. And this is going to be the value of the first term. That's 500 and 1 minus r. r is 1.03. And we raise that to the 7th power divided by 1 minus 1.03.
So to evaluate this fraction, first what we're going to do is apply the exponent to 1.03 so 1.03 to the 7th power. We subtract from 1. And this is going to be over 1 minus 1.03. That's negative 0.3. So cleaning up our fractions some more, we have a negative numerator and a negative denominator. So this is going to evaluate to a positive number.
So we're really multiplying $500 by 7.6625. And $500 multiplied by 7.6625 is $3,831.23. This represents a dollar amount. So my account balance after seven years is $3,831.23 with an initial deposit of $500, 3% interest gained each year, and an additional $500 deposited at the start of the year.
The next example has to do with the distance of an object. So an engineering student has designed this really neat rocket that shoots up in the air. But within constant intervals, it only travels one third of the distance previously traveled.
So after a certain time interval, it travels 20 feet in the air. And then after another time interval, it has only gone a third of that so 6.66 repeating. And after another time interval, it has traveled only a third of that. So that's 2.22 repeating. And then another time interval, it has only traveled a third of that previous distance 0.740 repeating.
So this is actually an infinite geometric sequence because we are continuing to cut the previous term into thirds. So it never actually reaches zero. But it gets very, very close to zero. It's an infinite geometric sequence.
And now, thinking about the sum of this infinite geometric sequence, this would be the height at which this rocket will eventually converge. So that means it's traveling to a distance is essentially zero. So it's going to be converging at a certain height.
So to find the sum of this infinite geometric sequence, we have a different formula. It's similar, but it's actually a little bit less complicated. We have the value of the first term. And we're multiplying it by 1 over 1 minus r. So the difference here is that we don't have an exponent in our numerator. It's just 1. We don't have the common ratio at all in our numerator either. It's just 1 over 1 minus r.
So using this formula then to find the sum of this infinite sequence, we have the sum as the value of the first term so that's 20 multiplied by 1 over 1 minus r. r is one third because it travels a third of the distance in each interval. So 1 minus 1/3 is 2/3.
So thinking about 1 divided by 2/3, this would be 1 multiplied by 3/2, the reciprocal. So 1 multiplied by 3/2, I can write as 1.5. And 20 times 1.5 is 30. So now, what does 30 represent? That represents feet. So this rocket will converge to a height of 30 feet. That's the sum of this infinite geometric sequence.
So let's review our tutorial on the sum of geometric sequences. Well, we found the sum of a finite geometric sequence using this formula here. And our example was with an investment account gaining interest. And we also made additional deposits to that account at the start of each year.
Our example with far infinite geometric sequence, the sum uses a different formula. And our example was with a rocket that converges to a certain distance, meaning that it travels a portion of the previous distance at constant intervals. So thanks for watching this tutorial on the sum of geometric sequences in real world situations. Hope to see you next time.
