Reccurence Relations

TenSaru SG

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 SG

A levels isit? use induction ba

wee_ws SG

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 SG

Our aim is to show that

U_(n + 1) - U_n > 0.

Very useful strategy and tactic

and very common

wee_ws SG

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