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 Show the first 4 terms, and then find the 8 th term. Use the explicit formula to write a geometric sequence whose common ratio is a decimal number between 0 and 1. Show the first four terms, and then find the 10 th term. Not a geometric sequence, we can still define Use the recursive formula to write a geometric sequence whose common ratio is an integer. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. Third term, so 4 times 3 times 2 times a. Term, and then my third one is going to be 3 times my second It was introduced in 1202 by Leonardo Fibonacci. The sequence shown in this example is a famous sequence called the Fibonacci sequence. 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 Seeing the pattern for an explicit formula for an arithmetic sequence or a geometric sequence will be easy as compared to finding explicit formulas for sequences that do not fall into these categories. 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 Armed with these summation formulas and techniques, we will begin to generate recursive formulas and closed formulas for other sequences with similar patterns and structures. Over there is a, ar to the 0 is just a, and then each Explicit formulas for geometric sequences. A geometric sequence is a sequence in which the ratio of any term to the previous term is constant. 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. Click on Open button to open and print to worksheet. Say a times r to the 2 minus 1, a times r to the first power. Worksheets are Geometric recursive and explicit work, Write the explicit formula for the, Recursive sequences, Unit 3c arithmetic sequences work 1, Notes 3, Arithmetic sequences date period, Using recursive rules with sequences, Given the following formulas find the first 4. 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 Using Recursive Formulas for Geometric Sequences A recursive formula allows us to find any term of a geometric sequence by using the previous term. Given a term in an arithmetic sequence and the common difference find the recursive formula and the three terms in the sequence after the last one given. 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. Where we start at some number, then each successive number \) so there is no common ratio.Geometric sequences, which is a class of sequences
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