SOLUTION: 1.the geometric mean between the first two terms is 32.if the third term is 4.find the first term

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Question 1185943: 1.the geometric mean between the first two terms is 32.if the third term is 4.find the first term
Found 3 solutions by MathLover1, ikleyn, greenestamps:
Answer by MathLover1(20849) About Me  (Show Source):
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Geometric Mean =32
a%5B3%5D=4
a%5B1%5D=?
Solution:
if Geometric Mean of a%5B1%5D and a%5B3%5D=4 is 32, we have
32=sqrt%28a%5B1%5D%2Aa%5B3%5D%29
32=sqrt%28a%5B1%5D%2A4%29
32=2sqrt%28a%5B1%5D%29
16=sqrt%28a%5B1%5D%29
a%5B1%5D=256

Answer by ikleyn(52780) About Me  (Show Source):
You can put this solution on YOUR website!
.
the geometric mean between the first two terms highlight%28of_a_geometric_progression%29 is 32.
if the third term is 4, find the first term.
~~~~~~~~~~~~~~~~~~~~~~~~


            @MathLover1 incorrectly read,  interpret and treat the problem.

            Her solution and her answer are incorrect.

            I came to bring a correct solution. 


The three terms of the geometric progression are  a, ar, ar^2,  from the first term to the third, inclusive.


Since the problem talks about "geometric mean", it means that the first two terms are POSITIVE,
and HENCE, all the terms of this GP are positive; in particular, "a" is positive and "r" is positive.


The geometric men of the first two terms is 

    GM(a,ar) = sqrt%28a%2A%28ar%29%29 = a%2Asqrt%28r%29.


So, the first equation is

    a%2Asqrt%28r%29 = 32.       (1)


The second equation is

    a%2Ar%5E2 = 4.        (2)


Dividing equation (2) by equation (1), you get

    r^(3/2) = 4%2F32 = 1%2F8 = 1%2F2%5E3.



Hence,  r^(1/2) = 1%2F2  and  r = 1%2F4.



THEREFORE,  the first term  is  a%5B3%5D%2Fr%5E2 = 4%2F%28%281%2F4%29%5E2%29 = 4%2F%28%281%2F16%29%29 = 4*16 = 64.



ANSWER.  The first term of the geometric progression is 64.


CHECK.   Then the terms of the progression are  64, 64%2A%281%2F4%29 = 16,  16%2A%281%2F4%29 = 4,

         and the geometric mean of the first two terms is  sqrt%2864%2A16%29 = sqrt%282%5E6%2A2%5E4%29%29 = sqrt%282%5E10%29 = 2%5E5 = 32.    ! Correct !

Solved   (correctly).



Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


The geometric mean between the first TWO terms is 32, and the third term is 4; from that we can conclude that the common ratio is less than 1.

Then, given that the geometric mean between the first two terms is 32, along with the fact that the common ratio is less than 1, we can do this:

Let the first term be 32/x
Let the second term be 32x

Then the first three terms of the geometric sequence are

32/x, 32x, 4

Now use the fact that the second term is the geometric mean between the first and third:

%2832x%29%5E2=%2832%2Fx%29%284%29
1024x%5E2=128%2Fx
x%5E3=128%2F1024=1%2F8
x=1%2F2

The common ratio is 1/2; the terms are
32/x = 32/(1/2) = 64
32x = 32(1/2) = 16
4

ANSWER: The first term is 64