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

It is given that `a_(1),a_(2),a_(3), . . .` are in H.P. Therefore, `(1)/(a_(1)),(1)/(a_(2)),(1)/(a_(3))` . . . Are in A.P. Let d be the common difference of the A.P.
`:." "(1)/(a_(n))=(1)/(a_(1))+(n-1)dand(1)/(a_(20))=(1)/(a_(1))+19d`
`rArr" "(1)/(a_(n))=(1)/(5)+(n-1)dand(1)/(25)=(1)/(5)+19d`
`rArr" "(1)/(a_(n))=(1)/(5)+(n-1)dandd=-(4)/(19xx25)`
`rArr" "(1)/(a_(n))=(1)/(5)-(4(n-1))/(19xx25)`
`rArr" "(1)/(a_(n))=(95-4n+4)/(19xx25)`
`rArr" "a_(n)=(99-4n)/(19xx25)`
Now, `a_(n)lt0`
`rArr" "(99-4n)/(19xx25)lt0`
`rArr" "99-4nlt0rArr4ngt99rArrngt24(3)/(4)rArrnge25`