Today we're going to talk about geometric sequences. A geometric sequence is just a set of numbers in numerical order with a nonzero common ratio between each term. So we're going to start by looking at a simple example of a geometric sequence. We'll look at some of the characteristics of the sequence. And then we'll do some examples solving problems involving geometric sequences.
So let's look at an example of a simple geometric sequence. I have 3, 12, 48, 192, and on and on. I know that a geometric sequence has a common ratio or a number that is being multiplied every time to get the next term. So to determine that number, I can either look at the sequence and see if I can identify the pattern-- I know that 3 times 4 is 12, and 12 times 4 is 48. So it seems that the pattern is multiplying by 4.
But if I couldn't see that or I wanted to verify, I could work backwards by dividing one term by the term in front of it. So 12 divided by 3 is 4. So I was right. The pattern is multiplying by 4, or the common ratio is 4. So I can continue that pattern of multiplying by 4 to find the next two terms. 192 multiplied by 4 is going to give me 768. And 768 multiplied by 4 is going to give me 3,072.
So now let's talk about divergent and convergent sequences. Our last example, 3, 12, 48, 192, and on and on is an example of a divergent sequence. And we call it a divergent sequence because as we continue to list out the terms, the values of the terms are heading towards infinity. And for a divergent sequence, the common ratio is such that the absolute value of the common ratio, or the absolute value of r, is going to be greater than 1.
For our convergent sequence, such as 10, 5, 2.51, 1.25, 0.25, et cetera, we can see that the values of our terms are getting smaller and tending towards 0. And we know that we're never going to have a negative value for term, because our common ratio is positive. So the terms are also stay positive. And in convergent sequences, as you continue to list out the terms, it's always going to tend towards a specific number. And the common ratio for a convergent sequence is such that the absolute value of r is going to be less than 1.
So when you're solving problems with a geometric sequence, it would take a long time and not be efficient to simply continue to multiply by a common ratio to find term in the sequence. So we can use a formula to find the value of any term in the sequence. And that formula is a sub n is equal to a sub 1 times r to the n minus 1.
So in this formula, an is the value of the nth term in the sequence. a1 then would be the value of the first term, or the initial term. r is going to be our common ratio, the number that we're multiplying by every time. And n is the number of the term in the sequence. So the first term, second term, third term-- that's our value for n. So we can use this formula to find the value of any term as well as solve other problems about the geometric sequence.
So let's take an example of a geometric sequence and write a formula that we can use to solve problems for that sequence. So I have a sequence, 25, 5, 1, 0.2, 0.04. I need to determine what my common ratio is. And I can do that simply by looking at the sequence. 25 times what number will give me 5? Or I could do that with any pair of numbers in the sequence. And if I'm not sure, I can work backwards by, again, picking any number in the sequence and dividing it by the term in front of it, the value of the term in front of it.
So I'm going to do this one. This one will be pretty easy. 0.2 divided by 1 is going to give me 0.2. So I know my common ratio is 0.2. Looking at my formula, I also need to know the value of the first term, a1. But I see that that's 25. So I'm ready to write my formula. So my formula for this sequence will be a sub n is equal to a1, 25, times r, 0.2, to the n minus 1. And now we can use this formula to solve problems for this geometric sequence.
So now let's do an example, solving a problem starting with the formula for a geometric sequence. I have an is equal to 0.25 times 2 to the n minus 1. So in this formula, 0.25 is our first term and 2 is our common ratio.
And here we want to find a5, which means we want to find the value of the fifth term. So using my formula, I'm going to start with a5 equals-- I"m substituting 5 here for n-- 0.25 times 2 to the 5 minus 1. I again am going to substitute 5 for n in the exponent.
Now I can simplify this expression. I'm going to start by simplifying my exponent. So this is just 0.25 times 2 to the 4th power. 2 to the 4th power gives me 16. So this is 0.25 times 16. And 0.25 times 16 is just going to be equal to 4. So I found that a5, or my fifth term in this sequence is 4.
So for my third example, I've got an is equal to 32. Find n. So we're using the same formula for the geometric sequence as my last example. But this time, I know that the value of a term in the sequence is 32. And I want to find n. I want to find what place in the sequence that 32 is.
So using my formula, if I know an is 32, I'm going to substitute 32 here for an. So I have 32 equals 0.25 times 2 to the n minus 1. To solve this, I have to isolate this n variable in the exponent. So I'm going to start by canceling out the 0.25 that's being multiplied in front. This will cancel. And 32 over 0.25 is 128. And I have 2 to the n minus 1 left.
Now, if I'm solving for an exponent, I need to use a logarithm. And now this equation is in exponential form. So I can convert this from exponential form to logarithmic form. So in logarithmic form, this becomes log-- my base is still 2-- of 128 is going to be equal to my exponent, n minus 1.
Now I'm going to use my change of base formula so that I can evaluate log base 2 of 128 using my calculator. So using the change of base formula, this is going to become log of 128 over log of 2. And that's still going to be equal to n minus 1.
So now that I've used my change of base formula, in my calculator, I can simply type in log of 128 over log of 2. And that's going to give me a value of 7. So 7 is equal to n minus 1. And then adding 1 on both sides, I find that n is equal to 8. So in my sequence, the 8th term is going to be 32.
So let's go over our key points from today. A geometric sequence is a set of numbers in numerical order with a nonzero common ratio between each term. To determine the common ratio, divide any term in the sequence by the term before it. And divergent sequences have a common ratio r such that the absolute value of r is greater than 1. Convergent sequences have a common ratio such that the absolute value of r is less than 1.
So I hope that these key points and examples helped you understand a little bit more about geometric sequences. Keep using your notes and keep on practicing, and soon you'll be a pro. Thanks for watching.
The ratio between any two consecutive terms in a geometric sequence; a constant value
A sequence whose terms have a finite limit; they tend toward a specific value
A sequence whose terms do not have a finite limit; they tend toward ± infinity
A set of numbers in numerical order, with a non-zero common ratio between each term