Just to get someĮxplicitly, but we can also define it recursively. ![]() Is not a geometric sequence, describes exactly this Me write this, this is 1, this is 2 times 1, thisĮqual to n factorial. Look at this particular, these particular The fourth one is essentially 4 factorial times a. Its set or it's the sequence a sub n from nĮquals 1 to infinity with a sub n being equal to, let's see Not a geometric sequence, we can still define Third term, so 4 times 3 times 2 times a. Term, and then my third one is going to be 3 times my second So this sequence that I justĬonstructed has the form, I have my first term,Īnd then my second term is going to be 2 times my first Here I'm multiplying itīy a different amount. Is this a geometric sequence? Well let's thinkĪbout what's going on. And then I could go to 120,Īnd I go on and on and on. Let me see what I want to do- I want to go to 24. So then I'm going to go to 2, and then I'm going to Now let me give youĪnother sequence, and tell me if it is geometric. So this could be 20 timesġ/2 is 10, 10 times 1/2 is 5, 5 times 1/2 isĢ.5- actually let me just write that as aįraction, is 5/2, 5/2 times 1/2 is 5/4, and you can just So the first term is 20,Īnd then each time we're multiplying by what? Well here each time Is 1, this is going to be 1/2 to the 0-th power. So what would this sequenceĪctually look like? Well let's think about it. And then r, theĮach successive term, let's say it's equal to 1/2. I could have a sub n, n isĮqual to 1 to infinity with, let's say, a sub n isĮqual to, let's say our first term is, I don't know, Successive term is going to be the previous Over there is a, ar to the 0 is just a, and then each Look, our first term is going to be a, that right ![]() Sub n minus 1, times r, for n is greater than or equal to 2. Making it very clear that a sub 1 is equal to a-Īnd then we could say a sub n is equal to the previous term, a Or we could say for n equalsġ, and then we could say a- and I don't even a sub 1 is equal to a,Īr to the 0 is just a. N equals 1 to infinity, with a sub 1 being equal to a. Say a times r to the 2 minus 1, a times r to the first power. Nth term is going to be ar to the n minus 1 power. This second term isĪr to the first power. To the zeroth power, r to the 0 is just 1. This right over here, a is the same thing as a times r The way to infinity, with a sub n equaling- well, Sequence is a sub n starting with the first term going all Ways we can denote it, we can denote it explicitly. Power, and you just keep on going like that. Multiply by rĪgain, you're going to get ar to the third I going to have? I'm going to have- it's aĭifferent shade of yellow- I'm going to have ar squared. Let's multiply it times,īut to get the third term, let's multiply the ![]() So what am I talking about? Well let's multiply a times r. Is the previous number multiplied by the same thing. Explanation: The recursive formula for the given geometric sequence is: an an-1 r. Where we start at some number, then each successive number Final answer: The recursive formula for the given geometric sequence is an a (n-1) r, where an represents the nth term in the sequence, a (n-1) represents the previous term, and r represents the common ratio. Maybe these having two levels of numbers to calculate the current number would imply that it would be some kind of quadratic function just as if I only had 1 level, it would be linear which is easier to calculate by hand.Geometric sequences, which is a class of sequences This gives us any number we want in the series. I do not know any good way to find out what the quadratic might be without doing a quadratic regression in the calculator, in the TI series, this is known as STAT, so plugging the original numbers in, I ended with the equation:į(x) = 17.5x^2 - 27.5x + 15. Then the second difference (60 - 25 = 35, 95-60 = 35, 130-95=35, 165-130 = 35) gives a second common difference, so we know that it is quadratic.
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |