Arithmetic and Geometric Progressions

Date: 10/10/95 at 5:15:49
From: Anonymous
Subject: Arithmetic/Geometric Progression

In an Arithmetic Progression whose first term and common 
difference areboth non-zero, U(n) denotes the nth term and S(n) 
denotes the sum of n terms. If U(6), U(4), U(10) form a Geometric 
Progression:

(i)   show that S(10) = 0
(ii)  show that S(6) + S(12) = 0
(iii) deduce that U(7) + U(8) + U(9) + U(10) = U(11) + U(12)

gsharpe@pioneer.as.edu.au

Date: 10/10/95 at 20:37:44
From: Doctor Ethan
Subject: Re: Arithmetic/Geometric Progression

Hey,

   Neat problem.  I will get you started then let you finish it. 
First we need some notation. I will use u4 instead of U(4) (to 
save use of the shift key)

Let u1=r
Let the common difference be a

Then u4 = 3a + r
u6 = 5a+r
u10 = 9a+r

Great. Then let B be the ratio.

So B=u4/u6=u10/u4

This will give us the equation  (3a + r)/(5a+r) = (9a+r)/(3a+r).

You can check that this solves to 4a(9a+2r)=0
But a can't be zero(see given) so it means that 9a + 2r = 0.,

Now see if you can use that to solve the problems. 
If not write back and I will do one for you.

-Doctor Ethan,  The Geometry Forum