Reccurence Relations

• TenSaru

  15 Mar 09, 23:46

  The sequence U1, U2, U3 ... is such that U(n+1)= (5Un + 4) / (Un + 2), U1=2

  Un=> 4 as n=>infinity (shown in previous part of question) ("=>" means approach)

   

  Given Un<4 for all positive integers n, show that Un<U(n+1) for all positive integers n...

  How to show? Must I draw a graph or something?

   

  Thanks in advance =)

• crimson soldier

  16 Mar 09, 04:17

  A levels isit? use induction ba

• wee_ws

  16 Mar 09, 07:50

  Hi, 

    Induction involving inequalities is not tested in the H2 mathematics syllabus.

    To show that U_n < U_(n + 1), consider U_(n + 1) - U_n

    and use the fact that U_(n + 1)= (5U_n + 4) / (U_n + 2). Our aim is to show that

    U_(n + 1) - U_n > 0.

    The resultant expression of U_(n + 1) - U_n will be an improper fraction which you will need to carry out long division in order to simplify it further. Then, you'll be able to see why it should be positive, given that U_n < 4.

    You may check out my website on Recurrence Relations at www.freewebs.com/weews

    Thanks in advance!

  Cheers,
  Wen Shih

• eagle

  16 Mar 09, 17:16

  Our aim is to show that

    U_(n + 1) - U_n > 0.

  Very useful strategy and tactic

  and very common

• wee_ws

  16 Mar 09, 19:58

  Hi,

    Another method to show that U_(n + 1) > U_n is to consider

    U_(n + 1) / U_n and show that it is greater than 1.

    Thanks!

  Cheers,
  Wen Shih