Which term of the series `sqrt2,2/3,(2sqrt2)/9,.....` is `16/2187`?

Options

A. 10th term
B. 8th term
C. 9th term
D. 11th term

Answer

B

Explanation

Given, series is
`sqrt(2),(2)/(sqrt(3))(2sqrt(2))/(9),…..`
or `(sqrt(2))/(3^(0)),((sqrt(2))^(2))/(3),((sqrt(2)^(3)))/(3^(2)),.....`
which is an infinite GP.
Whose, first term `a=(sqrt(2))/(3^(0))=sqrt(2)`
and common ratio
`=((sqrt(2)^(2)))/(3)xx(1)/(sqrt(2))=(sqrt(2))/(3)`
Let n term of GP `=(16)/(2187)`
`:. ar^(n-1)=(1)/(2187)`
`rArr sqrt(2)((sqrt(2))/(3))^(n-1)=(16)/(2187)`
`rArr ((sqrt(2))/(3))^(n-1)=(8sqrt(2))/(2187)=((sqrt(2))/(3))^(7)`
On comparing the powers. we get