SOLUTION: If the sums of the first three terms of an AP and a GP are equal and non-zero, the common difference of the AP and the common ratio of the GP are equal and the ratio of the AM to

Algebra.Com
Question 1136733: If the sums of the first three terms of an AP and a GP are equal and non-zero, the common difference of the AP and the common ratio of the GP are equal and the ratio of the AM to the GM is 1:-2, find the common ratio of the GP. Also, find the relationship between their first terms.
Found 2 solutions by greenestamps, ikleyn:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


The terms AM and GM are not, to my knowledge, commonly used; I don't know what they mean.

Please re-post your question.
Answer by ikleyn(52781)   (Show Source): You can put this solution on YOUR website!
.

            AM  is the standard abbreviation for  "Arithmetic Mean".

            GM  is the standard abbreviation for  "Geometric Mean".

            So, almost everything is clear,  except one thing:  When the condition says  "the ratio of the  AM  to the  GM  is  1:-2"
            it missed to say to which progressions these  AM  and  GM  do relate.

            So,  it requires some interpretation;  therefore I will re-edit the problem in this way: 

            The sums of the first three terms of an AP and a GP are equal  and non-zero,
            the common difference of the AP and the common ratio of the GP are equal, 
            and the ratio of the AM of AP to the GM of GP is 1:-2.
            Find the common ratio of the GP.

            by underlying my insertions.


Solution

Since this problem,  although elementary,  is a bit higher than the average School Math level,  I will assume that the reader's
level does correspond to the problem level,  and will not explain elementary details. 

Let " a " be the middle (the second) term of the AP  and " b " is the middle (the second) term of the GP.



Then it is widely known that AM of the first three terms of an AP is equal to "a", 
while the GM of the first three terms of an GP is equal to "b".

Therefore this statement of the condition  "the ratio of the AM of AP to the GM of GP is 1:-2"  simply means that  

     =  = .         (1)



Let "r" be the common difference of the AP and the common ratio of the GP (they are equal, so I use one symbol for both).

Then the fact that "The sums of the first three terms of an AP and a GP are equal"

gives the second equation

    3a =  + b + .       (2)


From equation (1), express  b = -2a  and substitute it into equation (2). You will get

    3a =  - 2a - (2a)*r.


Cancel factor "a" in both sides

    3  =  - 2 - 2r;

Multiply by "r" both sides

    3r = - 2 - 2r - 2r^2

and write the quadratic equation in the standard form

    2r^2 + 5r + 2 = 0.

Quadratic formula gives the roots  r = -2  and  r = .


So, the common difference of the AP may have these two values,

and the common difference of the GP may have these two values, RESPECTIVELY.


Having these solutions for "r", we can make one step forward and find  "the relationship between first terms of the AP and GP".



a)  If a = r = -2,  then   = a - r = a - 2;   =  = substitute here b = -2a from above =  = a.

    Therefore,   = a =  + 2  is that relationship between  and  in this case.



b)  If a = r = ,  then   = a - r = a + ;   =  = substitute here b = -2a from above =  = 4a.

    Therefore,   = 4a =  - 2  is that relationship between  and  in this case.


ANSWER.  There are two possibilities:  a)  r = -2  and   =  + 2;   and

                                       b)  r =   and   =  -2.

Solved.



RELATED QUESTIONS

The 2nd , 4th , and 7th terms of an AP are the first 3 consecutive terms of a GP . Find : (answered by MathLover1)
Find the difference between the sums to ten terms of the AP and GP whose first two terms (answered by ewatrrr)
The sum of first 20 and 50 terms of an AP is 420 and 2550. Find the eleventh term of a GP (answered by robertb)
The first, second and third term of a GP are the first, third and seventh term of an AP... (answered by greenestamps)
The sum of an AP is 4 and d first term is twice the third term of a GP,find (I)the 23rd... (answered by ikleyn)
The 100th term of an AP is 98, and the common difference is 2. Find the first three... (answered by richwmiller)
Two APs Have The Same First And Last Terms. The First AP Has 21 Terms With A Common... (answered by ikleyn)
(a)if log x , log(x+2) , log(x+16) are three consecutive terms of an AP, find the value... (answered by reviewermath)
The first and last terms of an AP are 67 and 171 respectively. If The first and last... (answered by Theo)