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Write the explicit and recursive formulas for 5,20,80,320...

Question

Write the explicit and recursive formulas for 5,20,80,320

Write the explicit and recursive formulas for 5,20,80,320



Answers

Write the explicit and recursive formulas for each geometric sequence. $$ 162,108,72,48, \dots $$

Okay on this problem, you're given the sequence 09 18 and 27 we're told that it's an arithmetic sequence and they want both the recursive and the explicit formula for this problem. So we know that the first number here is zero. So we need to find the common difference. So the common difference is two consecutive terms in the sequence subtracted. It should be identical. So nine minus zero, which gives me nine. So I know the common difference is fine. So the recursive formula is the first number. Ace of one is equal to zero. And then we have the rule says you take the first number with the previous number before it, and to the number would've redoing. Well, since it's arithmetic, we're not multiplying or providing. We're going to be adding nine to each number. And again, this is for when you're in is greater than or equal to two. The explicit formula for this problem is a seven, which means any number in the sequence. How do I find it? I'll take the first number when I add to it the common difference, which is nine animals. Why buy in minus one so we can simplify this. Um, we can distribute. So nine times in is nine in nine times. A minus one gives me minus nine. So my explicit formula for this one, and this again is for when you're in is greater than or equal to one.

Okay. We were given the geometric sequence of 36 12 and four when we're asked to write a recursive formula and an explosive formula for the sequence. So before we can write, either we need to find the common ratio. So to find the common ratio, you're going to take the second number divided by the first number in the sequence or term to six, divided by three, which is to. So we know that the common ratio for the given sequence is too. So the rights. Ah, recursive formula. We have to define the first numbers. That's your initial condition or initial seat. So the first number is three. The second part of the recursive formula is to say to generate the numbers in the sequence you take the previous number and again, I know in minus one, comes before in, and it says you keep multiplying by shoot, and this again is for ends that are greater than a religion to write the explicit formula, which allows you to go to any number of the sequence. So it's not based off of the previous number. We take the first number, which in this case again as three and we're gonna multiply by the R, which is to let me raise it to the and minus one, and that is for when in is greater than or equal to.

From here, we were asked to write recursive and the explicit formula for this given sequence. Now, to start off with, you need the same information you need to know your first term, and you'll need to know what you're comin. Ratio is. So to start off with, let's go with the recourse of formula. So the recursive formula we know are a one which stands for a first term. So if our first term here is at 27 we can automatically say a one is equal to 27. Now, for the second piece, we do need to know what are our is now are here. Santer are common ratio which could be found by taking your second term and dividing it by your first term. Because really, what you're looking for is what are we multiplying by each time to continue at sequins on. Now, if I took 36 divided by 27 we're gonna see that it is a decimal, or we can write as a fraction if we were to reduce it. So if I was to reduce that, I would see that we're going to get fourth or but you can also write as one point three repeating, or you can write it as one in 1/3. Either way would be correct. So are believe it as an improper fraction where employees, but in its for third. And we're gonna see it that is being multiplied by that a sub and minus one. Now, that whole thing there, then, is the recursive formula for this sequence. Now the second piece is writing the explicit. Now. Then I thought about it, is that you need the exact same information direct that formula, So not really a lot of extra work. It's just a matter of putting it in a different format. So we see that our first charm is still at 27. Can we see that our our is that 4 32 in a plug that in there as well? And that's being raised to the n minus one power. So that would be the explicit formula for the sequence as well. Did you have your recursive and your explicit, which is what that question is asking for

All right, take this. Found. They're asking us to really do two different things. There are enough to write, suited from Formula One being reverses and one being expensive. So the good thing about these problems are that you need the same information for both set of formula. So we need to know what first time and we need the O. R. Which is our common ratio. So for starters, her first term is always just the first number in our sequence. That 1 62 represents our first term. So I put that in for a one, Just a very first part of a repressive formula. I'm gonna copy that right over here. Just all lines up here at the end. Now, the other part is, we use in our common ratio. We need to know that our is the rest of the formula stays the same. All right. So the finer common ratio, which in this problem the numbers are decreasing in our sequence. That means we're multiplying by a fraction, which is sometimes not quite as obvious force. So honestly, to find our common ratio. If we just took our second term at one of late and divided by 1 52 are reduced It like a fraction they're gonna see that are common. Ratio is 2 30 meaning your mall signed by 2/3 each time to get through the next number in our secret. So we can take that 2/3 Plug that in to the second piece of the recursive formula. Now that is being multiplied by the A sub and minus one. We don't plug in anything for the end. We just simply write the formula, and that is the first piece. Now, the second piece being that explicit formula, we need the same information we needed in our first term. And we need you. Are, um we just discovered both of those things. That one should be pretty quick. Our first term being that 1 62 name a multi by our which we found it. We are 2/3 and that is being raised to the N minus one power again. We don't bug it. Anything for the end. So that is our solution.


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