Let,`a_1,a_2_a,a_3,….` be in harmonic progression with `a_1=5 " and " a_(20)=25` The least positive integer n for which `a_n lt 0 `

Options

A. 22
B. 23
C. 24
D. 25

Answer

D

Explanation

`:.a_(1),a_(2),a_(3),"......."` are in HP.
`:.(1)/(a_(1)),(1)/(a_(2)),(1)/(a_(3)),"......."`
Let D be the common difference of this AP, then
`(1)/(a_(20))=(1)/(a_(1))+(20-1)D`
`implies D((1)/(25)-(1)/(5))/(19)=-(4)/(25xx19)`
and `(1)/(a_(n))=(1)/(a_(1))+(n-1)D=(1)/(5)-(4(n-1))/(25xx19)=((95-4n+4)/(25xx19))`
`=((99-4n)/(25xx19))lt0" " [:.a_(n)lt0]`
`implies 99-4nlt0 implies ngt24.75`.