Let`a_(1),a_(2),"...."` be in AP and `q_(1),q_(2),"...."` be in GP. If `a_(1)=q_(1)=2 " and "a_(10)=q_(10)=3`, then

Options

A. `a_(7)q_(19)` is not an integer
B. `a_(19)q_(7)`is an integer
C. `a_(7)q_(19)=a_(19)q_(10)`
D. None of these

Answer

C

Explanation

`a_(1),a_(2),"....",` are in AP and `q_(1),q_(2),"....",` are in GP.
`a_(1)=q_(1)=2" and "a_(10)=q_(10)=3`
Let d be the common difference of AP
`i.e., d=(3-2)/(9)=(1)/(9)`
Then, `a_(7)=a_(1)+6d=2+6d=2+6xx(1)/(9)=(8)/(3)`
`a_(19)=a_(1)+18d=2+18d`
`=2+18xx(1)/(9)=(36)/(9)=4`
Let r be the common ratio of GP i.e.,`r=((3)/(2))^((1)/(9))`
Then, `q_(7)=q_(1)r^(6)=2r^(6)`b
`=2*((3)/(2))^(6xx(1)/(9))=2((3)/(2))^((2)/(3))`
`q_(10)=q_(1)r^(9)=2r^(9)=2*((3)/(2))^(9xx(1)/(9))=3`
`q_(19)=q_(1)*r^(18)=2r^(18)`
`=2*((3)/(2))^(18xx(1)/(9))=2((3)/(2))^((18)/(9))=(9)/(2)`
`a_(7)=q_(19)=(8)/(3)xx(9)/(2)=12,a_(19)=q_(10)=4xx3=12`