Let `a_(n)` be the `n^(th)` term of a G.P of positive integers. Let `sum_(n = 1)^(100) a _(2n) = alpha` and `sum_(n = 1)^(100) a_(2n +1) = beta` such that `alpha != beta`. Then the common ratio is

Options

A. `alpha//beta`
B. `beta//alpha`
C. `sqrt(alpha//beta)`
D. `sqrt(beta//alpha)`

Answer

A

Explanation

Let a be the first term and r be the common ratio of the given G.P. Then,
`alpha=sum_(n=1)^(100)a_(2n)`
`rArralpha=a_(2)+a_(4)+…+a_(200)`
`=ar+ar^(3)+…+ar^(199)`
`=ar(1+r^(2)+r^(4)+…+r^(198))`
`beta=sum_(n=1)^(100)a_(2n-1)`
`rArrbeta=a_(1)+a_(3)+...+a_(199)`
`=a+ar^(2)+..+ar^(198)`
`=a(1+r^(2)+....+r^(198))`
Clearly,`alpha//beta=r`.