Question Number 100624 by Dwaipayan Shikari last updated on 27/Jun/20

Commented by MJS last updated on 27/Jun/20

Commented by Dwaipayan Shikari last updated on 27/Jun/20

Commented by Dwaipayan Shikari last updated on 27/Jun/20

Answered by maths mind last updated on 27/Jun/20
![u_(n+1) ^2 =u_n +2...E u_0 =(√2) we can proof this withe logic 0≤u_n ≤2...easy too see u_(n+1) −u_n =((u_n +2−u_n ^2 )/( (√(u_n +2))+u_n ))=((−(u_n −2)(u_n +1))/( (√(u_n +2))+u_n ))≥0 u_n ↾ increasing ⇒ since0 ≤U_n ≤2 ⇒∃ (w_n ) such u_n =2cos(w_n ) proof w_n =cos^(−1) ((u_n /2)) well defind since (u_n /2)∈[0,1] since u_n ∈[0,2] and incrasing ⇒w_n ∈[0,(π/2)]⇒ w_n is deacrising E⇔4cos^2 (w_(n+1) )=2+2cos(w_n )⇒2cos^2 (w_(n+1) )−1=cos(w_n ) cos(2x)=2cos^2 (x)−1⇒ E⇔cos(2w_(n+1) )=cos(w_n ) ⇒2w_(n+1) =w_n because w_n ∈[0,(π/2)] ⇒w_(n+1) =(w_n /2)⇒w_n is geometric sequences ⇒w_n =w_0 ((1/2))^n w_0 =(π/4) u_n =2cos((π/(4.2^n )))=2cos((π/2^(n+2) ))](https://www.tinkutara.com/question/Q100635.png)
Commented by ghiniboss last updated on 28/Jun/20

Commented by prakash jain last updated on 28/Jun/20
