finding a recursive formula and its limit

How to find the limit from the explicit and also the recursive formula? The approach is very different!!

[meltossmedia]

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[elvinsamedov]

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[MrRyanroberson]

the implicit formula also allows you to generalize to all indexes, a_0 = 2, a_-1 = -1/2. The recursive formula agrees: a_0 = (-4-1)/(-9/2+2) = -10/(-9+4) = 10/5 = 2. a_1 = 15/20 = 3/4

[ffggddss]

Nice!

BTW, I did the recursive formula slightly differently, just because it was the first thing that occurred to me. In order to get a whole number I could pull out of the fraction, I went:
3a[n] = (3n+6)/(3n+1) = 1 + 5/(3n+1)

Then pick that apart one operation at a time, to this level:
3a[n] – 1 = 5/(3n+1)
5/(3a[n] – 1) = 3n + 1

Then subs. n+1 for n in the first line, and use what we just found:
3a[n+1] = 1 + 5/(3(n+1) + 1) = 1 + 5/(3n+1 + 3)
= 1 + 5/([5/(3a[n] – 1)] + 3)

Multiply top & bottom of the "big" fraction by (3a[n] – 1):
... = 1 + (15a[n] – 5)/(5 + 9a[n] – 3)
= 1 + (15a[n] – 5)/(9a[n] + 2)
= (24a[n] – 3)/(9a[n] + 2)

Finally, divide by 3:
a[n+1] = (8a[n] – 1)/(9a[n] + 2)

Essentially the same; just different turns onto different roads, to wind up at the same hotel. ;–)

Fred

[duggydo]

This is so cool! This is something we never covered in calculus classes. I feel like the teachers must have skipped over it.

[cameronspalding]

I’ve just realised: if we have a recursive relationship in the form a_n = f(a_n-1), f differential at L where L is a limit for a_n and |a_n-L| is of order 1/n^m where m>0 then
|f’(L)|=1

[josephhajj]

Your subscribers will diverge mr Blackpenredpen

[sergioh]

When you solve for L, isnt that coincidentally like solving for the auxiliary equation? And cool video btw as always! Very fun of course:D

[scarbotheblacksheep]

Circle around the plus? That's news to me.

[adam_salha]

Hey! I love your videos, you inspired me to study mathematics in college! I recently had an idea, if we have two fixed points A and B, and wanna see where C is such that angle ACB = 90° we can simply draw the circle of diameter [AB]. But what if the angle was different than 90°, how would the shape behave? If it's 30° for example or 45°!

[pierreabbat]

To check the recursive formula:
a=2; (8a-1)/(9a+2)=15/20=3/4.
a=5/10=1/2; (8a-1)/(9a+2)=3/6.5=6/13.

[jomama]

Hoping to see an analytic geometry vid on your channel :))

[andresxj]

So, for recursive formula, if the sequence converges, the limit doesn't depend on the initial value?

[damianmatma]

Thank you for this great video :)


I have a question related to your example:


Suppose we have given the recursive formula:
a.1 = 3/4
a.n = (8 * a_n-1 - 1) / (9 * a_n-1 + 2)


Now I want to get the explicit formula (we know we should get a.n = (n+2)/(3n+1) ).
Is it possible? If so, how to do that step-by-step?

It would be great if you could make another video (as Part 2 or as a continuation) with my problem solved.

[matheussales]

If I have the recursive formula A(n)=sqrt(A(n-1)+1), with A0=1, how do I deduce it's explicit form? Fun fact, the limit of this sequence when n goes to infinit is the golden ratio ^-^.

[Archik]

a0=2 recursive work for n=1 and give 3/4

[QoowedeDgikQmZG]

why don't use a_n - a_n-1 or a_n / a_n-1 to get the recursive function?

[mikeschieffer]

Nice video! Could you also show that asub(n-1) converges to 1/3 as n goes to infinity by using the intermediate equation you have of asub(n-1) = (n+1)/(3n-2)? Thanks for all you do!

[virendrasule]

6:42 but how does this method work if the recurrence of the sequence is not first order and also when an explicit expression for a_n is not available?

[jhonatanandres]

Excelent video, do you have any video about conical sections? (Special Problems, from graphic to general formula, the opposite...?)