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

Question

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

Ian Fowler · Accepted Answer

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.

Ian Fowler · Answer

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

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