ATAR Notes: Forum

VCE Stuff => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Mathematics => Topic started by: Jay.C on August 04, 2015, 02:06:18 pm

Title: Challenging Math.
Post by: Jay.C on August 04, 2015, 02:06:18 pm: Hey guys if you could answer the attached math question that would be amazing. I am asking on behalf of a much smarter friend. :)
Title: Re: Challenging Math.
Post by: keltingmeith on August 04, 2015, 06:54:02 pm: The subscripts are a little confusing. Reckon you could get your friend to confirm what they are?

(Also, is this actually from a VCE class? Because that seems quite above VCE level...)
Title: Re: Challenging Math.
Post by: Jay.C on August 04, 2015, 07:00:18 pm: Here is the photo of the question. Also no its not from a VCE class, I just wanted to see if any of you math geniuses on ATAR notes could help! ;D
Title: Re: Challenging Math.
Post by: wyzard on December 08, 2015, 05:05:14 pm: Quote from: Jay.C on August 04, 2015, 07:00:18 pm
Here is the photo of the question. Also no its not from a VCE class, I just wanted to see if any of you math geniuses on ATAR notes could help! ;D

This is a Real Analysis question under the topic sequences, and a really fierce one 8) I will not bog you down with the actual proofs which require the use of really advanced maths concepts; the definition of inferior and superior limit, mathematical induction and some theorems relating to sequences. The simplest explanation is that they are the "end points" of a set of numbers.

The inferior limit is 0, which can be shown by noting that every number in the sequence is greater than or equal to zero using mathematical induction, and since ${ a }_{ 1 }$ is zero, the inferior limit is 0.

The exterior limit on the other hand is 1, to show this we'll need to note 2 things. Firstly, it can be shown that ${ a }_{ 2k+1 }=\frac { { 2 }^{ k }-1 }{ { 2 }^{ k } }$ , so as $n$ goes to infinity, ${ a }_{ n }$ goes to 1 using the theorem $lim\quad { a }_{ 2k+1 }\quad =\quad lim\quad { a }_{ n }$ . Secondly it can be shown using mathematical induction that ${ a }_{ n }$ is definitely less than 1. Hence the exterior limit is 1.