blackpenredpen

math for fun 

  1. Discussions
  2. Math Problems
  3. Find the explicit formula of this sequence, if this is possible.
Search
Jotadiolyne Dicci
May 10, 2020

Find the explicit formula of this sequence, if this is possible.

Let U(n) be the sequence with n a natural number such that :

U(2n+1)-U(2n)=1

U(2n)-U(2n-1)=4

U(0)=3


Find the explicit formula of this sequence, if this is possible.


I honestly don't know how to begin.

Except that it's fairly easy to see that:

- when n is even, U(n) = U(n-1)+4

- when n is odd, U(n)=U(n-1)+1

0
2
Ian Fowler
  ·  May 10, 2020  ·  Edited: May 10, 2020

For n even: U(n) = 5n/2 + 3

For n odd: U(n) = 5n/2 + 3/2

I just wrote out the first 8 terms

3,4,8,9,13,14,18,19,... and separated the odd and even positions

3,8,13,18,..

4,9,13,19,...


Both are arithmetic with common difference = 5 and I just picked the constant to start in the right place.


You can also add your 2 recursion formulas to get: U(2n+1) - U(2n-1) = 5 so

U(2n+1) = 5 + U(2n-1). i.e. go back 2 and add 5 and that helped me get the 5n/2


Jotadiolyne Dicci
May 10, 2020

Oh i didn't think in that way thanks !

0
Ian Fowler
May 10, 2020

@Jotadiolyne Dicci You are very welcome

0
Jotadiolyne Dicci
May 10, 2020

@Ian Fowler However, is that possible to make just one formula without the for n even and n odd which brings together everything ?

0
Load more replies
2
Ian Fowler
May 11, 2020  ·  Edited: May 11, 2020

Sorry for the small size but you can blow it up by clicking the image. It took a long time but I finally got my head in the right place. I tried to use Generating Functions , which works for Fibonacci, but not so well here. At any rate, I got it !!


I tried to explain as I went along but feel free to ask more questions. What a great problem - I learned a lot about how to apply alternating sequences along the way. Thanks very much.


BPRP are you out there?? IMHO this definitely deserves a video.




Jotadiolyne Dicci
May 11, 2020

Good use of the (-1)^n. I didn't see it ! It's easier than using cos !

0
1
Ian Fowler
May 11, 2020

I made a slip in (I): s(n) = (-1)^n should read 1,-1,1,-1,1,-1,.... not 1,0,1,0,1,0, ....