Recursive formula for geometric sequences
About this video
Given the terms 51, 153, 459, 1,377, the video notices each term is 3 times the one before, so the sequence is geometric. It writes the recursive formula as f(n) = 3 f(n - 1) and a_n = 3 a_(n-1), and finds the initial (zeroth) term by dividing 51 by 3 to get f(0) = a_0 = 17.
You'll be able to
- tell whether a sequence grows by adding or by multiplying the same amount.
- write a recursive formula for a geometric sequence in function and subscript notation.
- work backwards to find an initial term by dividing by the common ratio.
Key words
- recursive formula
- A rule that gives each term using the term before it, plus a starting term.
- geometric sequence
- A sequence where you multiply by the same number to get each next term.
- arithmetic sequence
- A sequence where you add or subtract the same amount each time.
- initial term
- The starting term of a sequence, here f(0) or a_0.
- subscript notation
- Writing terms as a_n, where the small n shows the term's position.
Transcript
- [Instructor] We're told the terms one through four of a sequence are shown below. Write the recursive formula for the sequence in two forms, including the initial term. So either we can use the form with the function notation, and the initial term would be F of zero, or we could use, I would use the term subscript notation, A subzero.
So pause this video, have a go at this before we do it together. So let's just start looking at what's going on here. So this is the first term, this is the second term, third term, and then fourth term here. That's a little confusing with that comma, but this is 1,377.
And before this, we're going to have a zeroth term that we're going to want to figure out. But when we go from 51 to 153, well, first we have to figure out is this an arithmetic sequence? Are we adding or subtracting the same amount, or is that the difference changing, which means that we're probably multiplying, not always the case, but maybe multiplying by the same amount.
Well, the difference seems to be expanding, and when you look at it, it looks like to go from 51 to 153, you multiply by three, you multiply by three again, and yes, you get to 459, and you multiply by three again, I'm doing this roughly in my, yeah, that looks about right.
So it looks like what is happening here is every term, so F sub N, the Nth term is equal to the term right before that times three. So we could write three times F of N minus one. Or if we wanted to use the subscript notation here, I could write A sub N, The Nth term is equal to three times the term before it, the N minus first term.
Now, what is F of zero or A sub zero? Well, to go one term back, instead of multiplying by three, we just divide by three. Or another way of thinking about it is what do we have to multiply by three to get to 51? Well, 51 divided by three is 17, so we could write F of zero is equal to 17.
That's that initial condition, so to speak, or where we're starting from, or the initial term. Or we could write A of zero, A sub zero is equal to 17, and we're done.
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