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Find a function $f(n)$ that identifies the $n$ th term $a_{n}$ of the following recursively defined sequences, as $a_{n}=f(n)$.$a_{1}=2$ and $a_{n+1}=2 a_{n}$ for $...

Question

Find a function $f(n)$ that identifies the $n$ th term $a_{n}$ of the following recursively defined sequences, as $a_{n}=f(n)$.$a_{1}=2$ and $a_{n+1}=2 a_{n}$ for $n geq 1$

Find a function $f(n)$ that identifies the $n$ th term $a_{n}$ of the following recursively defined sequences, as $a_{n}=f(n)$.$a_{1}=2$ and $a_{n+1}=2 a_{n}$ for $n geq 1$



Answers

Find an explicit formula for the $n$ th term of the sequence satisfying $a_{1}=0$ and $a_{n}=2 a_{n-1}+1$ for $n \geq 2.$

All right, So this problem deals with one of the most quintessential examples of the sequence. So we know that the first term anyone was one and a and plus first term, because minus a So they give you a recursive lee to find sequence and the first term. So you're given the whole secret. And this is one of those very quintessential examples to weaken right out the first several terms. Do we have one negative one one negative form for? So it seems quite quite apparent with going on, you're reversing poles and we could use a tricking a metric function to deal with this, but I think it's even easier than that. So well, we are looking to identify in terms of bed. So for the end of the 1st 1 that should be one. And what's really helpful to do here is you just have the negative one and and like that because and it's even, uh, we wanted to end. Why this one? Yeah. And last one. So it would get zero. So one. And then when any equals to be negative one ankles, 31 and so on versa. And we'll be even this fact we'll be even during either during odds. So and there poor

Anyone caught you one And then we have the from love funder A and plus one. You go to the inn plus one times with the end for the and critically go to one. And then I tried to find a Jew. Ico June Uh ah, This were weak. Jud Hamm's no previous term. And then you could you do and then a three echo to three times the previous term to hear and equal to six a five far in with equal Judah far times the 16 coaches a gente for a five echo Judah. Five times that he was term tune If Mico Judah seven. Okay, so five times, Gent, my attempts gently flopped you quote, you know, 120 and inch and so on. He noticed them that I want we can understand, you know one factorial to equal to Juve Toyo six Equal territory for Theriot For Jennifer Coggiola Far Vittorio five Pretoria Their father A and must ico Judah in Victoria

I e one equal to chew. And then we have a general term endless one Nico to the endless one times a nd value. But you and for the end, greater it could you one here we can try to find a ate you equal itude two times their previous term too. Even made you so get equal to the chew 83 equal to the three times their previous term Defending bite. You gonna go to three a far ico June The farm attempts their previous time development you and we get equal. Jude Ah ah six a five echo to five times debris was term depending but you. Then we get equal to the fit deal and so on. Here. Notice that we can write the A one in We go to the one factorial on then and dividing by Ah ah half ah half We can register into the to power off the ah one minus two and then similarly this one equal to the two factorial dividing by two power off the two wants to hear we have when we go to the three. Fraternal If I didn't mind the to about three minus one and they will hand off off a Toyota dividing by Jew. About four minutes. One if I'm honest you here. Sorry. And this one recorded a five fraternal development. You power bi minus two. So, in general, held any cogent and Victorian, if any, May 2 hour and minus two.

The angle on equal to one on da A and less one ego to I am defending But you about and for any greater equity one. So let's try to find the and what I do it we go June the rivers term one The bunion binder to our once we got your half and then a three echo to reverse time one half, depending by Jew about you. So we get this one we could, you know, one over to power three isn't Cuban one here and is about far ego. If I called you one on one to about three development, you about three. They were getting one over to power six, and then a five equals to one on June 6 over the Jew power far. So we get in court. You want to, uh, 10. And so they're noticed that we can write the I want in code to the one driven by ju power zero. Here we can register into one over to power off the ah to temp's one day even invite you. And here we can get this going to be one over through about. I'm not three times to living by Do you again. The one over Cuba phone time Street devalued by two and so on. So doesn't have to, uh, five times for defending by two. So in general, we have a n must equity one of, ah, two hour and minus one times and the venue. But you


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