Question Number 100640 by pticantor last updated on 27/Jun/20

let( U_n ) be a sequence definied by:   { ((U_0 =1)),((U_(n+1) =((3U_n +2)/(U_n +2)))) :}  show that 0<U_n <2

Answered by maths mind last updated on 27/Jun/20

U_n ≥0  U_(n+1) =2+((u_n −2)/(u_n +2))    u_n =1≤2    by recurence  supose  0≤u_n ≤2  ⇒((u_n −2)/(u_n +2))≤0⇒u_(n+1) =2+((u_n −2)/(u_n +2))≤2

Commented bypticantor last updated on 28/Jun/20

thank you sir

Answered by 1549442205 last updated on 28/Jun/20

using induction method:  +for n=0,u_0 =1∈(0;2)⇒state is true  +suppose that the state is true for n=k  which means that u_k =((3u_(k−1) +2)/(u_(k−1) +2))∈(0;2)  +Consider u_(k+1) =((3u_k +2)/(u_k +2)).Clearly,u_(k+1) >0  u_(k+1) <2⇔((3u_k +2)/(u_k +2))<2⇔3u_k +2<2(u_k +2)  ⇔u_k <2 .But this inequality is true by  induction hypothesis ,so the inequality    u_(k+1) <2 is true which show that  the state is also true for n=k+1.  Hence,by induction principle it is true  for ∀n∈N(q.e.d)  Furthermore,we can prove above sequence  increasing and Lim_(n→∞) U_n =2

Commented bypticantor last updated on 28/Jun/20

thank you sir