Răspuns:
52
Explicație pas cu pas:
m(∡A₁OAₙ₊₁)=69°20'=69+(20/60)=69+(1/3)=(69·3+1)/3=208/3.
m(∡A₁OAₙ₊₁)=m(∡A₁OA₂)+m(∡A₂OA₃)+m(∡A₃OA₄)+...+m(∡AₙOAₙ₊₁)=
[tex]=\dfrac{1}{3}*140+\dfrac{1}{15}*140+\dfrac{1}{35}*140+...+\dfrac{1}{(2n-1)(2n+1)}*140=\\=140*(\dfrac{1}{3}+\dfrac{1}{15}+\dfrac{1}{35}+...+\dfrac{1}{(2n-1)(2n+1)})=\\=70*(\dfrac{2}{1*3}+\dfrac{2}{3*5}+\dfrac{2}{5*7}+...+\dfrac{2}{(2n-1)(2n+1)})=\\=70*(\dfrac{1}{1}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{2n-1}-\dfrac{1}{2n+1})=\\ 70*(\dfrac{1}{1}-\dfrac{1}{2n+1})=70*(\dfrac{2n+1}{2n+1}-\dfrac{1}{2n+1})=70*\dfrac{2n+1-1}{2n+1}=70*\dfrac{2n}{2n+1} \\[/tex]
[tex]Deci,~70*\dfrac{2n}{2n+1}=\dfrac{208}{3},~=>~[/tex] 420n=208(2n+1), ⇒420n=416n+208, ⇒420n-416n=208, ⇒4n=208, ⇒n=52
