My Math Forum  

Go Back   My Math Forum > High School Math Forum > Algebra

Algebra Pre-Algebra and Basic Algebra Math Forum


Thanks Tree3Thanks
  • 1 Post By Math Message Board tutor
  • 1 Post By soroban
  • 1 Post By v8archie
Reply
 
LinkBack Thread Tools Display Modes
April 16th, 2015, 11:44 AM   #1
Senior Member
 
Joined: Jan 2015
From: USA

Posts: 107
Thanks: 2

Sequence problem

Please help me with this problem Im having a hard time with sequences and series!
Attached Images
File Type: jpg Sequence.jpg (18.1 KB, 10 views)
matisolla is offline  
 
April 16th, 2015, 03:14 PM   #2
Banned Camp
 
Joined: Jun 2014
From: Earth

Posts: 945
Thanks: 191

Quote:

$\displaystyle A \ \ number \ \ sequence \ $ $\displaystyle \ \{a_n\}, \ \ \ (n = 1, \ 2, \ 3, \ ...) \ \ satisfies \ \ the \ \ following \ \ conditions.$

Express $\displaystyle \ \{a_n\} \ $ as a function of $\displaystyle \ n.$

$\displaystyle 3a_{n + 1} \ = \ 2a_n \ + \ 1, \ \ \ (n = 1, \ 2, \ 3, \ ...), \ \ \ \ a_1 = 2.$
$\displaystyle a_1 = 2$

$\displaystyle 3a_2 = 2(2) + 1 = 5$

$\displaystyle 3a_2 = 5$

$\displaystyle a_2 = \dfrac{5}{3}$

After this, $\displaystyle \ \ \dfrac{5}{3}$ will take on the role that $\displaystyle \ a_1 = 2 \ $ had above.


Continue this process.


Including $\displaystyle \ a_1 = 2, \ $ the first five numbers are:


2, 5/3, 13/9, 35/27, 97/81


I found this sequence difficult to determine a pattern, but I wrote them as this eventually:


$\displaystyle \dfrac{2}{1}, \ \dfrac{5}{3}, \ \dfrac{13}{9}, \ \dfrac{35}{27}, \ \dfrac{97}{81} \ = $


$\displaystyle \dfrac{ 1 + 1}{1}, \ \dfrac{3 + 2}{3}, \ \dfrac{9 + 4}{9}, \ \dfrac{27 + 8}{27}, \ \dfrac{81 + 16}{81} $


Can you come up with a general formula for $\displaystyle \ a_n \ $ in terms of $\displaystyle \ n \ $ by studying that?
Thanks from matisolla
Math Message Board tutor is offline  
April 16th, 2015, 05:37 PM   #3
Math Team
 
Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
Thanks: 408

Hello, matisolla!

Quote:
5) A number sequence satisfies the following conditions:



Express as a function of



























Thanks from matisolla
soroban is offline  
May 24th, 2015, 05:17 AM   #4
Senior Member
 
Joined: Jan 2015
From: USA

Posts: 107
Thanks: 2

Quote:
Originally Posted by soroban View Post
Hello, matisolla!































Thanx Soroban! This method looks very useful but I cant understand the theory behind it. Does this method of solving sequences have a name? I would like to read further about it.

By the way, are you Japanese? Because I speak Japanese and "Soroban" is a very fitting name for someone like you lol
matisolla is offline  
May 24th, 2015, 10:14 AM   #5
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 7,634
Thanks: 2620

Math Focus: Mainly analysis and algebra
Generally, a difference equation has solutions $A\alpha^n$ where $\alpha$ is a root of the characteristic polynomial (quadratic in the case above). The general solution is a linear combination of each of the terms (one for each root).

One can also derive a solution using the first order equation. Suppose that $a_n = f(n)$ is a solution of $3a_{n+1}-2a_n= 1$ and that $a_n = g(n)$ is a solution of the homogeneous equation $3a_{n+1}-2a_n= 0$, Then $a_n = f(n) + cg(n)$ is also a solution of the nonhomogeneous equation (where $c$ is a constant).

Thus we may seek a constant solution to the nonhomogeneous equation and add to it some multiple of the solution to the homogeneous equation. Our initial condition will fix the multiple.

Thus we seek a constant $a$ such that $3a - 2a = 1$ which implies $a=1$, and the solution of the nonhomogeneous equation $a_{n+1}={2 \over 3}a_n$ which is $g(n) = \left( {2 \over 3} \right)^n$.

Thus our general solution is $a_n = 1 + c\left( {2 \over 3} \right)^n$ where $c$ is a constant determined by the initial condition.
Thanks from matisolla
v8archie is offline  
Reply

  My Math Forum > High School Math Forum > Algebra

Tags
problem, sequence



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Sequence, Difficult problem baku Algebra 7 December 1st, 2013 06:27 PM
Sequence problem Akcope Number Theory 1 November 7th, 2013 12:28 PM
Problem of sequence Snowflakes Calculus 2 April 4th, 2012 11:53 AM
another sequence problem?? bedii Real Analysis 2 October 12th, 2009 03:29 AM
sequence convergence problem peka0027 Real Analysis 4 March 25th, 2009 05:45 PM





Copyright © 2019 My Math Forum. All rights reserved.