Home
Class 12
MATHS
A sequence of positive terms A(1),A(2),A...

A sequence 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

Text Solution

Verified by Experts

`:.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)]`
Promotional Banner

Similar Questions

Explore conceptually related problems

8,A_(1),A_(2),A_(3),24.

If A_(n) is the set of first n primes then bignn_(n=3)^(10) A_(n)=

The number of ways in which ten candidates A_(1), A_(2), .. .., A_(10) can he ranked, if A_(1) is always above A_(2) is

If 1, a_(1), a_(2). ..., a_(n-1) are n roots of unity, then the value of (1 - a_(1)) (1 - a_(2)) .... (1 - a_(n-1)) is :

If a_(1),a_(2),a_(3),".....",a_(n) are in HP, than prove that a_(1)a_(2)+a_(2)a_(3)+a_(3)a_(4)+"....."+a_(n-1)a_(n)=(n-1)a_(1)a_(n)

If a_(1) , a_(2) , a_(3),"………."a_(n) are in H.P. , then a_(1), a_(2) + a_(2) a_(3) + "………" + a_(n-1)a_(n) will be equal to :

Let a_(1), a_(2) , 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 is :

The number of all possible triples (a_(1),a_(2),a_(3)) such that a_(1) + a_(2)cos2x + a_(3)sin^(2) x = 0 for all x is :

Let a_(1),a_(2),a_(3),"......"a_(10) are in GP with a_(51)=25 and sum_(i=1)^(101)a_(i)=125 " than the value of " sum_(i=1)^(101)((1)/(a_(i))) equals.

Find the sixth term of the sequence a_(n) =(n)/(n+1) ?