`a_(1), a_(2), a_(3),`…………. are distinct terms of an A.P. We cal `(p,q,r)` an increasing triad if `a_(p),a_(q),a_(r)` are in G.P. where `p,q,repsilonN` such that `pltqltr`. If `(5,9,16)` is an increasing triad, which of the following option is/are correct

Options

A. if `a_(1)` is a multiple of 4 then every term of the A.P. is an integer
B. `(85,149,261)` is an increasing triad
C. if the common difference of the A.P. is `1/4`,then its first term is `1/3`
D. ratio of the `(4k+1)th` term and `4k^(th)` term can be 4

Answer

A, B, C

Explanation

Let `R` be the common ratio of the G.P. and `D` be the common difference of A.P.
`a_(5)=a_(5), a_(9)=Ra_(5), a_(16=a_(5)R^(2)`
`a_(9)-a_(5)=4Dimplies(R-1)a_(5)=4D`…..(1)
`a_(16)-a_(9)=7DimpliesR(R-1)a_(5)=7D`………(2)
from equation `(1)//(2)` we get `1/r=4/7=R=7/4`
From equation `(2)-(1)` we get `(R-1)^(2)a_(5)=3D=(8a_(5))/16=3D`
`3/16(a_(1)=4D)=Dimplies3/16 a_(1)=(1- 3/4)Dimpliesa_(1)=(4D)/3`