Let `a_n` be the nth term of a G.P. of positive numbers. 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
(a) `alpha//beta` b. `beta//alpha` c. `sqrt(alpha//beta)` d. `sqrt(beta//alpha)`

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=underset(n=1)overset(100)suma_(2n)=a_(2)+a_(4)+ . . . +a_(200)=ar^(3)+ . . . .ar^(199)`
`rArr" "alpha=ar(1+r^(2)+r^(4)+ . . .+r^(198))`
`and,beta=underset(n=1)overset(100)suma_(2n-1)=a_(1)+a_(3)+ . . .+a_(199)=a+ar^(2)+ . . .ar^(198)`
`rArr" "beta=a(1+r^(2)+ . . .+r^(198))`
Clearly, `alpha//beta=r`.