Question
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?
Answers
anonymous 2.0
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
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