Consider the sequences (u_n ) and (v_n ) defined by { ((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 (1) Show that (u_n ) and (v_n ) are strictly positive also Show that (u_n ) and (v_n ) are of opposite sense of variation. (2) let w_n = v_n −u_n show that 0 ≤ w_(n+1) ≤ (1/2)w_n (3) Prove by induction that 0 ≤ w


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Question Number 100769 by Rio Michael last updated on 28/Jun/20
$Consider the sequences (u_n ) and (v_n ) defined by { ((u_0 = 1)),((u_(n+1) = ((2u_n v_n )/(u_n + v_n )))) :} and { ((v_0 = 2)),((v_(n+1) = ((u_n + v_n )/2))) :} ∀ n∈ N (1) Show that (u_n ) and (v_n ) are strictly positive also Show that (u_n ) and (v_n ) are of opposite sense of variation. (2) let w_n = v_n −u_n show that 0 ≤ w_(n+1) ≤ (1/2)w_n (3) Prove by induction that 0 ≤ w_n ≤ (1/2^n )$
$Consider the sequences (u_{n}) and (v_{n}) defined by$ ${\begin{cases} u_{0} = 1 \\ u_{n + 1} = \frac{2 u_{n} v_{n}}{u_{n} + v_{n}} \end{cases} and {\begin{cases} v_{0} = 2 \\ v_{n + 1} = \frac{u_{n} + v_{n}}{2} \end{cases} \forall n \in N$ $(1) Show that (u_{n}) and (v_{n}) are strictly positive also$ $Show that (u_{n}) and (v_{n}) are of opposite sense of variation .$ $(2) let w_{n} = v_{n} - u_{n} show that 0 ⩽ w_{n + 1} ⩽ \frac{1}{2} w_{n}$ $(3) Prove by induction that 0 ⩽ w_{n} ⩽ \frac{1}{2^{n}}$
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