Three positive numbers form an increasing G.P. If the middle term in this G.P. Is doubled , the new numbers are in A.P. Then the common ratio of the G.P. Is?
If the first is a, then the 2nd is ar
In the AP, then,
2ar-a = ar^2-2ar
r^2 - 4r + 1 = 0
r = 2±√3
Since we know r>0, r=2+√3
check:
without loss of generality, w can take a=1. Then we have
GP: 1, 2+√3, 7+4√3
if we double the 2nd term, we have
1, 4+2√3, 7+4√3
and the common difference is 3+2√3
To find the common ratio of the given geometric progression (G.P.), we can use the given information about doubling the middle term.
Let us assume the three positive numbers in the G.P. are a/r, a, and ar, where "a" is the first term and "r" is the common ratio.
According to the question, if we double the middle term, we get a new arithmetic progression (A.P.). The new numbers are a/r, 2a, and ar.
In an arithmetic progression, the common difference (d) between consecutive terms remains the same. So, we can find the common difference by subtracting consecutive terms in the new A.P.
The common difference, d, in the new A.P. can be found as follows:
2a - a/r = ar - 2a
To simplify the equation, we multiply through by r:
2ar - a = a - 2ar
Now, let's continue simplifying:
3ar = 2a
Divide both sides by a:
3r = 2
Divide both sides by 3:
r = 2/3
Therefore, the common ratio of the given geometric progression (G.P.) is 2/3.