Recursive formula for arithmetic sequence
About this video
Given terms three to six of a sequence (-3, -9/4, -3/2, -3/4), the video rewrites them all in fourths to see that each term adds 3/4. It writes the recursive formula as f(n) = f(n - 1) + 3/4 and a_n = a_(n-1) + 3/4, then works backwards to find the first term, f(1) = a_1 = -9/2.
You'll be able to
- find the common difference of an arithmetic sequence by using a common denominator.
- write a recursive formula in function notation and subscript notation.
- work backwards from a later term to find the first term.
Key words
- arithmetic sequence
- A sequence where you add or subtract the same amount each time.
- recursive formula
- A rule that gives each term using the term before it, plus a starting term.
- common difference
- The amount added each time in an arithmetic sequence; here it is 3/4.
- subscript notation
- Writing terms as a_1, a_2, a_n, where the small number tells the position.
Transcript
- [Instructor] We're told terms three through six of a sequence are shown below, here they are. Write the recursive formula for the sequence in two forms, including the first term and the first term would either be f of one or a sub one. And so when they're saying the two forms, one is using function notation and one is using kind of this lower case with subscript notations.
So pause this video, have a go at that before we do this together. Okay, now let's do this together and let's just think about what's happening when we go from one term to the next term. So if we go from negative three to negative nine fourths, so we don't know if this is an arithmetic or a geometric sequence, so we don't know if we're multiplying by the same thing every time or we're adding or subtracting by the same thing every time.
But when I look at this, it looks like we are decreasing by the same amount. Let's see, if I were to write all of these with the same denominator, this would be negative, I'll do four. So this would be negative 12 fourths, and then we have negative, then we have negative nine over four.
Then we have negative six over four, I'm just converting that into fourths. And then we have negative three over four. So when you write it this way, it looks pretty clear that we are decreasing by three fourths every time. You take three fourths, oh, sorry, no, we are increasing by three fourths.
Because we started at negative 12 fourths, you add three fours, you get to negative nine fourths. So every time we are adding three fourths. So I can even, I can start to write the recursive formula. We haven't figured out what the first terms are yet, but I could write f of n.
The nth term is going to be equal to the term right before that. So the f of n minus one, the n minus oneth, f of n minus oneth term and then plus three fourths. Or if I want to use this notation, I could write a sub n is equal to a sub n minus one. The term right before that plus three fourths.
So now let's figure out what our initial terms are. So we already said, Hey, this right over here is the third term. So we could even call that a sub three if we want, we could also call that f of three. So if we were to go in the other direction, we're going to subtract three fourths every time.
So the term right before this, if I subtract three fourths, I'm going to get negative 15 over four. That is a sub two. And then if I subtract three fourths again, I get negative 18 over four. That is a sub one or also going to be equal to f of one. Now negative 18 over four is the same thing as negative nine halves.
So we could write f of one is equal to negative nine halves. Or over here we could say a sub one is equal to negative nine halves. And we're done.
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