recursive formula for a geometric sequence

Notice that each number in the sequence is the sum of the two numbers that precede it.

And the most classic recursive formula is the Fibonacci sequence. The exact same results.Staircase Analogy Recursive Formulas For SequencesĪlright, so as we’ve just noted, a recursive sequence is a sequence in which terms are defined using one or more previous terms along with an initial condition. Well what's g of 2? Well g of 2, we alreadyįigured out is 9 times 8. Let me scroll overĪ little bit so I don't get all scrunched up. And of course this wasĮquivalent to g of 2. So this is going toīe equal to 9 times 8. And what's g of 1? Well g of 1 is right over here. To g of 2 minus 1 times 8, which is the same thingĪs g of 1 times 8. What happens when x equals 2? Well when x equals 2, thisĬase doesn't apply anymore. Of 1- well if x equals 1, it's equal to 9. But we'll see that itĪctually does work out. Saying hey, g of x, well if x doesn't equal 1 it's going Why it's recursive is it's referring to itself. Use this recursive definition for g of x. Going to have x and we're going to have g of x. X other than 1, g of x is equal to the previousĮntry- so it's g of- I'll do that in a blue color. That this one right over here is going to be the previousĮntry, g of x minus 1. You multiply the 8s and we have a 9 in front,

Here is g of x minus 1, however many times Let's go all the way down to x minus 1, and then an x. So that took care ofĮquals anything else it equals the previous g of x. So we could sayīecause I'm overusing the red. So let's define that asĪ recursive function. When x equals 1 is 9, every term after that is 8 When x is 4, this is going to beĨ to the 4 minus 1 power, or 9 to the 3rd power. I think you see a littleīit of a pattern forming. So we could write thatĪs 9 times 8 times 8. Then when x is equal toģ, what's going on here? Well this is going So that's the same thing asĩ times 8 to the 1st power. And then see what theĬorresponding g of x is. And let's thinkĪbout what happens when we put in various x's To understand the inputs and outputs here. Write a recursive definition of this exact same function. So this is an explicitlyĭefined function. So we could say theĭomain of this function, or all the valid inputs X being a positive- or if x is a positive- integer. Of x is equal to 9 times 8 to the x minus 1 power. Would I be right to make these conclusions?Īlso interested to hear any feedback on the recursive formula. Explicit ones are a lot simpler and faster. It would make sense, as I can't really see the value of recursive formulas if explicit ones always work. It seems that for a formula on the given issue to work, it is necessary to refer back to the previous term, and that you can't make explicit formulas for any sequence, even if you can make a recursive one.

I've been trying for over an hour now though, to convert this into an explicit one, with no luck. It seems I've managed to make a recursive one that works: I was inspired by these formulas to make an explicit and recursive formula for the remaining debt after payment per term on a mortgage, with a fixed payment per term plan. Feel free to ask any questions for further explanation. I know that is a mouthful, but it is about the best way I can explain it. So in mathematical terms this means 4 * (-0.5)^n-1 times. Since f(1) is always 4 this means you can say 4 * -0.5 (whatever your n is, minus one, and you have to multiply times 0.5 this many times. so like f(50), that means you have to multiply f(1) * -0.5 49 times. So now you should see a pattern that whatever n equals you have to multiply f(1) n-1 times -0.5 to get the answer. to get f(4) you had to multiply f(1) * -0.5 three times. to get f(3) you had to multiply f(1) * -0.5 two times. But if you notice to get f(2) you had to multiply f(1) * -0.5 one time. then you could use f(2) to work out f(3) till you got to f(50). since you know f(1)=4 because it was given to you, you can plug in the numbers. You would have to get 47, then 46, then 45 until you got all the way back to 1. This is why it is called 'recursive', the answer needs you to get the answer of the previous term to get your final answer. So you would have to find f(49) So you plug it in. Well then you need to know what f(49) is. If someone were to ask you 'what is f(50).

Recursive formula for a geometric sequence