a 1 = 9 a n + 1 = 5 + a n 2 5 < a n ≤ 6 {\displaystyle {\begin{aligned}&a_{1}=9\\&a_{n+1}={\frac {5+a_{n}}{2}}\\&5<a_{n}\leq 6\\\end{aligned}}}
1. a 1 = 9 {\displaystyle \ a_{1}=9} אינו בתחום 5 < a n ≤ 6 {\displaystyle \ 5<a_{n}\leq 6} 2. a n + 1 = 5 + a n 2 a 1 + 1 = 5 + a 1 2 a 2 = 5 + 9 2 = 7 {\displaystyle {\begin{aligned}&a_{n+1}={\frac {5+a_{n}}{2}}\\&a_{1+1}={\frac {5+a_{1}}{2}}\\&a_{2}={\frac {5+9}{2}}=7\\\end{aligned}}} אינו בתחום 5 < a n ≤ 6 {\displaystyle \ 5<a_{n}\leq 6} 3. a n + 1 = 5 + a n 2 a 2 + 1 = 5 + a 2 2 a 3 = 5 + 7 2 = 6 {\displaystyle {\begin{aligned}&a_{n+1}={\frac {5+a_{n}}{2}}\\&a_{2+1}={\frac {5+a_{2}}{2}}\\&a_{3}={\frac {5+7}{2}}=6\\\end{aligned}}} בתחום 5 < a n ≤ 6 {\displaystyle \ 5<a_{n}\leq 6}
5 < a k < 6 {\displaystyle \ 5<a_{k}<6}
5 < a k + 1 < 6 {\displaystyle {\begin{aligned}&5<a_{k+1}<6\\\end{aligned}}}
a k + 1 = 5 + a k 2 a k 46 + 70 k − 24 k 46 2 ( a k + 1 ) = 5 + a k a k = 2 a ) k + 1 − 5 {\displaystyle {\begin{aligned}&a_{k+1}={\frac {5+a_{k}}{2}}\\&{\frac {ak}{46}}+{\frac {70^{k}-24^{k}}{46}}\\&2(a_{k+1})=5+a_{k}\\&a_{k}=2a){k+1}-5\\\end{aligned}}}
נציב את a k {\displaystyle \ a_{k}} ב 5 < 2 a k + 1 − 5 < 6 {\displaystyle \ 5<2a_{k+1}-5<6} ונקבל 5 < 2 a k + 1 − 5 ≤ 6 {\displaystyle 5<2a_{k+1}-5\leq 6}
5 < 2 a k + 1 − 5 10 < 2 a k + 1 2.5 < 2 a k + 1 {\displaystyle {\begin{aligned}&5<2a_{k+1}-5\\&10<2a_{k+1}\\&2.5<2a_{k+1}\\\end{aligned}}}
2 a k + 1 − 5 < 6 2 a k + 1 < 11 a k + 1 < 5.5 {\displaystyle {\begin{aligned}&2a_{k+1}-5<6\\&2a_{k+1}<11\\&a_{k+1}<5.5\\\end{aligned}}}
2.5 < 2 a k + 1 − 5 ≤ 5.5 {\displaystyle \ 2.5<2a_{k+1}-5\leq 5.5} ז"א 5 < a k + 1 ≤ 6 {\displaystyle \ 5<a_{k+1}\leq 6}
