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Question Number 103260 by 675480065 last updated on 13/Jul/20

	Answered by Rio Michael last updated on 13/Jul/20
	${ ((u_0 = 1)),((u_(n+1) = ((2u_n v_n )/(u_n + v_n )))) :} and { ((v_0 = 2)),((v_(n+1 ) = ((u_n + v_n )/2))) :} ∀ n ∈ N (a) n ∈ N ⇒ 2u_n v_n > 0 andu_n + v_n > 0 since u_n and v_n > 0 hence u_n and v_n are strickly positive sequences. (b) w_n = v_n −u_n ⇒ w_(n+1) = v_(n+1) −u_(n+1) ⇒ w_(n+1) = ((u_n + v_n )/2)−((2u_n v_n )/(u_n + v_n )) = ((u_n ^2 + 2u_n v_n + v_n ^2 −4u_n v_n )/(2(u_n + v_n ))) = ((u_n ^2 −2u_n v_n + v_n ^2 )/(2(u_n + v_n ))) ⇒ w_(n+1) = (((u_n −v_n )^2 )/(2(u_(n ) + v_n ))) ⇒ w_(n+1) ≤ (1/2)(v_n −u_n ) = (1/2)w_n hence 0 ≤ w_(n+1) ≤ (1/2)w_n (c)lim_(n→∞) w_n = 0$
	${\begin{cases} u_{0} = 1 \\ u_{n + 1} = \frac{2 u_{n} v_{n}}{u_{n} + v_{n}} \end{cases} and {\begin{cases} v_{0} = 2 \\ v_{n + 1} = \frac{u_{n} + v_{n}}{2} \end{cases} \forall n \in N$ $(a) n \in N \Rightarrow 2 u_{n} v_{n} > 0 and u_{n} + v_{n} > 0 since u_{n} and v_{n} > 0$ $hence u_{n} and v_{n} are strickly positive sequences .$ $(b) w_{n} = v_{n} - u_{n} \Rightarrow w_{n + 1} = v_{n + 1} - u_{n + 1}$ $\Rightarrow w_{n + 1} = \frac{u_{n} + v_{n}}{2} - \frac{2 u_{n} v_{n}}{u_{n} + v_{n}} = \frac{u_{n}^{2} + 2 u_{n} v_{n} + v_{n}^{2} - 4 u_{n} v_{n}}{2 (u_{n} + v_{n})} = \frac{u_{n}^{2} - 2 u_{n} v_{n} + v_{n}^{2}}{2 (u_{n} + v_{n})}$ $\Rightarrow w_{n + 1} = \frac{{(u_{n} - v_{n})}^{2}}{2 (u_{n} + v_{n})} \Rightarrow w_{n + 1} ⩽ \frac{1}{2} (v_{n} - u_{n}) = \frac{1}{2} w_{n}$ $hence 0 ⩽ w_{n + 1} ⩽ \frac{1}{2} w_{n}$ $(c) \lim_{n \to \infty} w_{n} = 0$
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