Using recursive formulas of geometric sequences | Mathematics I | High School Math | Khan Academy

The geometric sequence ( a_i ) is defined by the formula where the first term ( a_1 ) is equal to -1/8 and then every term after that is defined as being so ( a_i ) is going to be two times the term before that. So, ( a_i ) is ( 2 \times a_{i-1} ).

What is ( a_4 ), the fourth term in the sequence? Pause the video and see if you can work this out.

Well, there's a couple of ways that you could tackle this. One is to just directly use these formulas. So, we could say that ( a_4 ), well, that's going to be this case right over here: ( a_4 ) is going to be equal to ( 2 \times a_3 ).

Well, ( a_3 ) if we go and use this formula is going to be equal to ( 2 \times a_2 ). Each term is equal to two times the term before it, and then we can go back to this formula again and say ( a_2 ) is going to be ( 2 \times a_1 ): ( 2 \times a_1 ).

And lucky for us, we know that ( a_1 ) is -1/8, so it's going to be ( 2 \times -1/8 ), which is equal to -1/4.

So, this is -1/4. Thus, ( 2 \times -1/4 ) is equal to -1/2 or -1/2.

So, ( a_4 ) is ( 2 \times a_3 ). ( a_3 ) is -1/2, so this is going to be ( 2 \times -1/2 ), which is going to be equal to -1.

So that's one way to solve it. Another way to think about it is: look, we have our initial term and we also know our common ratio. We know each successive term is two times the term before it.

So, we could explicitly write it as ( a_i ) is going to be equal to our initial term -1/8, and then we're going to multiply it by two. We're going to multiply it by ( 2^{i-1} ) times.

So, we could say, times ( 2^{i-1} ). Let's make sure that makes sense. So ( a_1 ) based on this formula ( a_1 ) would be -1/8 times ( 2^{1-1} ) which is ( 2^0 ). So that makes sense; that would be -1/8 based on this formula.

( a_2 ) would be -1/8 times ( 2^{2-1} ) so ( 2^1 ). So we're going to take our initial term and multiply it by 2 once, which is exactly right.

( a_2 ) is -1/4. So if we want to find the fourth term in the sequence, we could just say, well, using this explicit formula, we could say ( a_4 ) is equal to -1/8 times ( 2^{4-1} ), which is -1/8 times ( 2^3 ).

So this is equal to -1/8 times 8.

Thus, -1/8 times 8 equals -1.

And you might be a little bit tossed up on which method you want to use, but for sure this second method right over here where we come up with an explicit formula once we know the initial term and we know the common ratio—this would be way easier if you were trying to find, say, the 40th term.

Because doing the 40th term recursively like this would take a lot of time and frankly a lot of paper.