My Math Forum  

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

Elementary Math Fractions, Percentages, Word Problems, Equations, Inequations, Factorization, Expansion


Thanks Tree8Thanks
  • 1 Post By mathman
  • 2 Post By greg1313
  • 2 Post By Hoempa
  • 1 Post By greg1313
  • 2 Post By aurel5
Reply
 
LinkBack Thread Tools Display Modes
September 17th, 2015, 08:35 AM   #1
Newbie
 
Joined: Sep 2015
From: Kenya

Posts: 6
Thanks: 0

Question Arithmetical and Geometrical Progression

The 2nd, 5th and 14th term in an arithmetical progression form the 1st three terms in a Geometrical Progression. The 10th term in the Arithmetical progression is 57. Find the :

a) 1st term and common difference in the AP
b) common ratio in the GP

HELP PLEASE
Worryinglamb08 is offline  
 
September 17th, 2015, 01:19 PM   #2
Global Moderator
 
Joined: May 2007

Posts: 6,732
Thanks: 689

Quote:
Originally Posted by Worryinglamb08 View Post
The 2nd, 5th and 14th term in an arithmetical progression form the 1st three terms in a Geometrical Progression. The 10th term in the Arithmetical progression is 57. Find the :

a) 1st term and common difference in the AP
b) common ratio in the GP

HELP PLEASE
Let a = AP first term, i = AP increment
Let c = GP first term, m = GP multiplier

Your equations:
a+i=c
a+4i=cm
$\displaystyle a+13i=cm^2$
a+9i=57

You now have 4 equations in 4 unknowns.
Thanks from Worryinglamb08
mathman is offline  
September 17th, 2015, 01:49 PM   #3
Senior Member
 
aurel5's Avatar
 
Joined: Apr 2014
From: Europa

Posts: 575
Thanks: 176

We have the AP:

$\displaystyle a_1,\ a_2,\ a_3, ...$

If the common difference is d, then

$\displaystyle a_2=a_1+d
\\\;\\
a_5=a_1+4d
\\\;\\
a_{14}=a_1+13d$

We have the GP:

$\displaystyle a_1+d,\ a_1+4d,\ a_1+13d,\ ...

$

These three consecutive terms will satisfy the equation:

$\displaystyle (a_1+4d)^2=(a_1+d)(a_1+13d) \ \Longrightarrow\ ...\ \Longrightarrow\ d = 2a_1\ \ \ (*)

$

In the AP, we know that:

$\displaystyle a_{10} = 57 \ \Longrightarrow\ a_1+9d=57 \ \stackrel{(*)}{\Longrightarrow}\ a_1+9\cdot2a_1=57 \ \Longrightarrow\ 19a_1=57 \ \Longrightarrow\ a_1=3.
\\\;\\
a_1=3\ \stackrel{(*)}{\Longrightarrow} d= 6.$
aurel5 is offline  
September 17th, 2015, 03:12 PM   #4
Math Team
 
Joined: Apr 2010

Posts: 2,780
Thanks: 361

As 2 + 2 * 14 = 3 * 10, $\displaystyle a_2 + a_{14} + a_{14} = 3 \cdot a_{10} = 171$,

171 = a2 * (2r^2 + 1) where r is the common ratio. (14 - 5) / (5 - 2) = 3 so r = 3 giving $\displaystyle a_2 = 9$ so $\displaystyle a_{14} = 81$ giving d = (81 - 9) / (14 - 2) = 6.
Hoempa is offline  
September 17th, 2015, 03:55 PM   #5
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,932
Thanks: 1127

Math Focus: Elementary mathematics and beyond
$\displaystyle a_1=57-9d$

$\displaystyle \dfrac{a_1+4d}{a_1+d}=\dfrac{a_1+13d}{a_1+4d}$

Substitute for $\displaystyle a_1$ and solve for $\displaystyle d$.
Thanks from Hoempa and Worryinglamb08
greg1313 is offline  
September 18th, 2015, 02:26 AM   #6
Math Team
 
Joined: Apr 2010

Posts: 2,780
Thanks: 361

Greg, that shows two solutions!
Thanks from greg1313 and Worryinglamb08
Hoempa is offline  
September 18th, 2015, 07:35 AM   #7
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,932
Thanks: 1127

Math Focus: Elementary mathematics and beyond
Yes, one of which is always $\displaystyle d=0$.
Thanks from Worryinglamb08
greg1313 is offline  
September 18th, 2015, 11:17 AM   #8
Senior Member
 
aurel5's Avatar
 
Joined: Apr 2014
From: Europa

Posts: 575
Thanks: 176

Quote:
Originally Posted by greg1313 View Post
Yes, one of which is always $\displaystyle d=0$.


If d = 0, we have a beautiful and gentle banality:

$\displaystyle a_1=a_2=a_3= ... =a_{10}= ...= a_n = 57

$

So, we all celebrate another reality.
Thanks from greg1313 and Worryinglamb08
aurel5 is offline  
Reply

  My Math Forum > High School Math Forum > Elementary Math

Tags
arithmetical, geometrical, progression



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Averages of arithmetical functions raul21 Number Theory 7 June 10th, 2014 01:41 PM
Arithmetic Progression and Geometric Progression jiasyuen Algebra 1 May 1st, 2014 03:40 AM
Updated asin arithmetical function PKSpark Computer Science 7 February 28th, 2014 02:16 AM
Help on these unknown arithmetical methods mirror Number Theory 3 September 7th, 2013 08:21 AM
Van der Wearden Theorem about arithmetical progression teodork Number Theory 1 December 10th, 2007 02:24 PM





Copyright © 2019 My Math Forum. All rights reserved.