Question 41934: Find four geometric means between 4096 and 972.
Thank you
Found 2 solutions by fractalier, AnlytcPhil:
Answer by fractalier(6550)   (Show Source):
You can put this solution on YOUR website! You divide 972 by 4096 and then take the fifth root to find the common ratio r...since you need four intermediate geometric means, there will be five increments...when you do that you find that r = 3/4...so that your sequence looks like...
4096, 3072, 2304, 1728, 1296, 972
Answer by AnlytcPhil(1806)  (Show Source):
You can put this solution on YOUR website! Find four geometric means between 4096 and 972.
This means to make a geometric sequence with 6 terms with
the 1st term being 4096, and the 5th term 972, and four
terms (geometric means) between them. In other words,
you are to find the missing four numbers in
4096, ____, ____, ____, ____, 972
so that the six terms will be a geometric sequence.
We use the formula
an = a1rn-1
with n=6
a6 = a1r6-1
a6 = a1r5
Now we substitute a6 = 972, and a1 = 4096
972 = 4096r5
972/4096 = r5
243/1024 = r5
35/45 = r5
(3/4)5 = r5
Take fifth roots of both sides
3/4 = r
So we multiply the 1st term, a1, which is 4096,
by r, which is 3/4, and get
a2 = 4096(3/4) = 3072
Then we multiply the 2nd term, a2, which is 3072,
by r, which is 3/4, and get
a3 = 3072(3/4) = 2304
Then we multiply the 3rd term, a3, which is 2304,
by r, which is 3/4, and get
a4 = 2304(3/4) = 1728
Then we multiply the 4th term, a4, which is 1728,
by r, which is 3/4, and get
a5 = 1728(3/4) = 1296
Then finally, as a check, we multiply the 5th term, a5,
which is 1296, to see if we get the 6th term, which is
given as 972, by r, which is 3/4, and see if we get
972.
a6 = 1296(3/4) = 972
Yes we do, so the geometric sequence is
4096, 3072, 2304, 1728, 1296, 972
and the four geometric means between the 1st and 6th terms are
the four terms between them:
3072, 2304, 1728, and 1296
Edwin
AnlytcPhil@aol.com
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