# 3 rational numbers form a GP, if 8 is added to the middle number, then this new sequence forms an AP...

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3 rational numbers form a geometric progression in its current order.

1. If 8 is added to the middle number, then this new sequence forms an arithmetic progression.
2. If 64 is later added to the last number, then this yet another sequence forms a geometric progression again.
3. First number of these sequences is an integer.

Find the numbers?

posted May 10, 2017

Let a,b,c be the numbers in GP.
Then we have the geometric mean = (a+c)^(0.5) = b
Then the terms become a, b+8, c which is in AP
Then arithmetic mean = (a+c)/2 = b+8.
Then the terms become a, b+8, c+64
Then geometric mean = (a+(c+64))^(0.5) = b+8

Now we have the solution as

a, (a + ((64a)^0.5/2)), (a + ((64a)^0.5/2))^(2)/a
Using trial and error and some common sense (Lol) for a=4 we get
a=4, b=12, c=36 which is in GP with 3 as the common ratio
Then
a=4, b=20, c=36 which is in AP with 16 as the common difference.
Then
a=4, b= 20, c=100 which is again in GP with 5 as the common ratio.

So a=4, b=12 & c=36 is a solution for the question.

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