A squence of positive terms `A_(1),A_(2),A_(3),"....,"A_(n)` satisfirs the relation `A_(n+1)=(3(1+A_(n)))/((3+A_(n)))`. Least integeral value of `A_(1)` for which the sequence is decreasing can be

Explanation

`:.A_(n+1)=(3(1+A_(n)))/((3+A_(n)))" For "n=1, A_(2)=(3(1+A_(1)))/((3+A_(1)))`
For `n=2,A_(3)=(3(1+A_(2)))/((3+A_(2)))`
`=(3(1+(3(1+A_(1)))/((3+A_(1)))))/(3+(3(1+A_(1)))/((3+A_(1))))=(6+4A_(1))/(4+2A_(1))=(3+2A_(1))/(2+A_(1))`
`:.` Given, sequence ccan be written as
`A_(1),(3(1+A_(1)))/((3+A_(1))),((3+2A_(1)))/((2+A_(1)))"...."`
Given,`A_(1)gt0` and sequence is decreasing, then
`A_(1)gt(3(1+A_(1)))/((3+A_(1))),((3+A_(1)))/((3+A_(1)))gt((3+2A_(1)))/((2+A_(1)))`
`implies A_(1)^(2)gt3 " or "A_(1)gtsqrt(3)`
`:. A_(1)=2 " " [" least integral value of " A_(1)]`