n' arithmetic means are there between 4 and 36. If the ratio of 3rd and (n-2)th mean is 2:3, find the value of n.

Answer

N/a

Explanation

Let 'n' arithmetic means between 4 and 36 be `A_(1),A_(2),…,A_(n).`
`therefore" "4,A_(1),A_(2),...,A_(n),"36 are in A.P."`
`"and "T_(n+2)=36`
`rArr" "4+(n+2-1)d=36`
`rArr" "(n+1)d=32`
`rArr" "d=(32)/(n+1)" "...(1)`
`"Given that "(A_(3))/(A_(n-2))=(2)/(3)`
`rArr" "T_(4)/(T_(n-1))=(2)/(3)`
`rArr" "(4+3d)/(4+(n-2)d)=(2)/(3)`
`rArr" "8+(2n-4)d=12+9d`
`rArr" "(2n-13)d=4`
`rArr" "(2n-13).(32)/(n+1)=4" [From eq. (1)]"`
`rArr" "64n-416=4n+4`
`rArr" "60n=420`
`rArr" "n=7.`