three consecutive terms of a geometric progression are the 1st,2nd and 6th terms of an ap.find the common ratio

Answer 1

first
a = a
a r = a + d
a r^2 = a + 5 d
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a (r-1) = d so a =d/(r-1)
d r^2 / (r-1) = d/(r-1) + 5 d
d r^2 = d + 5d (r-1)
r^2 = 1 + 5 r - 5
r^2 - 5 r +4 = 0
(r - 4)(r-1) = 0
r = 1 is not much of a series so
r = 4