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Sum of a Finite Geometric Sequence

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Today we're going to talk about finding the sum of a finite geometric sequence. Remember, a geometric sequence is just a set of numbers in numerical order that has a common ratio between each term. So the pattern is multiplying by a number to get to the next term. So we'll do some examples, finding the sum of a finite geometric sequence using a formula, and by simply adding the terms.

So let's do an example using the formula for the sum of a finite geometric sequence. I've got our formula here. In this formula, sn is the sum of the n terms. a1 is the first term in the sequence. r is our common ratio between any two consecutive terms in the sequence, and n is the number of terms that we're summing.

So for example, if I have the sequence 110, 1, 0.1, I can find the sum of these four terms. So since I'm finding the sum of four terms, my number for n, my value for n is 4. a1 is 100, that's my first term in the sequence. r I can find by dividing a term in the sequence by the one before it. So I'm going to do 0.1 divided by 1, because I know that's just 0.1.

My value for n is still 4. And in the denominator, I have 1 minus 0.1. So now simplifying, I'm going to start by simplifying my exponent in the numerator, 0.1 to the fourth power gives me 0.0001. Bringing down the rest of my terms, I'm going to subtract in the numerator. That's going to give me 0.9999. 0.9 in the denominator. And dividing those two terms, that gives me 1.111.

Now finally, I can multiply 100 times 1.111. And they give me a value of 111.1 for the sum of my four terms. And we can verify that by simply adding the four terms in the sequence. So 100 plus 10 is going to give me 110. If I add 1 to that, that's going to give me 111. And if I add 0.1 to that, that's going to give me a 111.1, which is what I found using my formula.

So let's do a second example with a different geometric sequence finding the sum. So this time, I've got the sequence 2, 3, 4.5, 6.75. So let's start by finding the sum using our formula. So we again are going to find the sum of the four terms. So my value for n is 4, a1 is 2, 1 minus my common ratio here is going to be 1.5. So I can see that I'm multiplying by 1.5 every time to get the next term.

My value for n is again 4. I'm going to simplify, starting in the numerator with my exponent, 1.5 to the 4th power is 5.0625. Now when I subtract 1 minus 5.0625, this is going to give me a negative number, negative 4.0625. And I know that my sum is going to need to be positive. However, because the denominator of my fraction is also going to be negative, negative 0.5, when I divide, the value of this ratio is going to be positive, which will make my sum a positive value.

So dividing these two, I get 8.125. So this is 2 times 8.125. And multiplying those two values, I find the sum of the first four terms, or the four terms in the sequence, is 16.25. So let's see if we can verify that by just summing the four terms. So 2 plus 3 gives me 5. 5 plus 4.5 is going to give me 9.5. And 9.5 plus 6.75 does gives me 16.25.

So for this example, I've got the geometric sequence 7, negative 7, 7, negative 7. I'm going to find the sum of the four terms in this sequence using our formula. So I know that my value for n is going to be 4. I have four terms that I'm summing. My value for a1 is 7. My value for r, the common ratio, I can find by taking any term in the sequence and dividing it by the term in front of it. So negative 7 divided by 7 is negative 1. So r is negative 1.

My value for n is 4 again. And in the denominator, I'll have 1 minus negative 1. So I'm going to start simplifying this in my numerator, with the exponent negative 1 to the 4th power as positive 1. Then I'll subtract, in my numerator and in my denominator, 1 minus 1 in the numerator is 0. 1 minus negative 1 in the denominator is going to give me 2. 0 divided by 2 is just 0. And 7 times 0 is going to give me 0. So I've found that the sum of these four terms is going to be 0.

And we could see that, by looking at the sequences, it's a pretty simple sequence. A positive and a negative number added together is always going to give you 0. So when we add another one, that will give us 0 again. So the sum of this finite geometric sequence is 0.

So finally, let's look and see how we can use the formula to find the partial sum of a sequence. So for example, I have the geometric sequence 2, 1, 0.5, 0.25. I can find the sum of just part of the sequence. So instead of all four terms, I just want to find the sum of three of the terms. And we could find the partial sum of any geometric sequence, whether it's finite like this one, or infinite.

So I'm going to start by using my formula to find the sum of these three terms. So now, even though I have four terms in my sequence, I only want to find the sum of three of them. So my value for n is 3. My first term is not going to be the first term that's listed in the sequence, but the first term that I'm trying to sum. So 1.

Then I'll have 1 minus-- my value for r is 0.5. I can see that I'm multiplying by 0.5 every time to get the next term. My value for n is still 3. And my denominator is 1 minus 0.5. Now I'm going to simplify, again, starting by in the numerator with my exponent, 0.5 to the 3rd power is 0.125. 1 minus 0.125 is going to give me 0.875. And 1 minus 0.5 in the denominator is 0.5. 0.875 divided by 0.5 is 1.75, and 1 times 1.75 is 1.75.

So I've found that the sum of those three terms is 1.75, and I can verify that by simply summing these three terms. In this case, it's pretty easy. 1 plus 0.5 is 1.5, plus the 0.25 is going to give me 1.75.

So let's go over our key points from today. In the formula for the sum of a finite geometric sequence, a1 is the first term in the sequence, r is the common ratio between consecutive terms, and n is the number of terms. The formula for finding the sum of a finite geometric sequence can be used when r is both positive and negative. And the formula can also be used to calculate a partial sum of a finite or infinite sequence.

So I hope that these key points and examples helped you understand a little bit more about sum of a geometric sequence. Keep using your notes, and keep on practicing, and soon you'll be a pro. Thanks for watching.

Formulas to Know

- Sum of a Finite Geometric Sequence