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September 17th, 2015, 08:35 AM  #1 
Newbie Joined: Sep 2015 From: Kenya Posts: 6 Thanks: 0  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 
September 17th, 2015, 01:19 PM  #2  
Global Moderator Joined: May 2007 Posts: 6,732 Thanks: 689  Quote:
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.  
September 17th, 2015, 01:49 PM  #3 
Senior Member 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: $\displaystyle a_{10} = 57 \ \Longrightarrow\ a_1+9d=57 \ \stackrel{(*)}{\Longrightarrow}\ a_1+9\cdot2a_1=57 \ \Longrightarrow\ 19a_1=57 \ \Longrightarrow\ a_1=3. \\\;\\ a_1=3\ \stackrel{(*)}{\Longrightarrow} d= 6.$ 
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. 
September 17th, 2015, 03:55 PM  #5 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,932 Thanks: 1127 Math Focus: Elementary mathematics and beyond 
$\displaystyle a_1=579d$ $\displaystyle \dfrac{a_1+4d}{a_1+d}=\dfrac{a_1+13d}{a_1+4d}$ Substitute for $\displaystyle a_1$ and solve for $\displaystyle d$. 
September 18th, 2015, 02:26 AM  #6 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361 
Greg, that shows two solutions!

September 18th, 2015, 07:35 AM  #7 
Global Moderator 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 $\displaystyle d=0$.

September 18th, 2015, 11:17 AM  #8 
Senior Member Joined: Apr 2014 From: Europa Posts: 575 Thanks: 176  

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arithmetical, geometrical, progression 
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