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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 |
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Answer» `:.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)]` |
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