a n + 1 = a n + 8 n a 1 = 1 a n = ( 2 n − 1 ) 2 {\displaystyle {\begin{aligned}&\color {green}a_{n+1}=a_{n}+8n\\&a_{1}=1\\&\color {blue}a_{n}=(2n-1)^{2}\end{aligned}}}
a n = ( 2 n − 1 ) 2 = ( 2 − 1 ) 2 = 1 a 1 = 1 1 = 1 √ {\displaystyle {\begin{aligned}&a_{n}=(2n-1)^{2}=(2-1)^{2}=1\\&a_{1}=1\\1=1\surd \\\end{aligned}}}
a k = ( 2 k − 1 ) 2 {\displaystyle \ a_{k}=(2k-1)^{2}}
a k + 1 = ( 2 k + 1 ) 2 {\displaystyle {\begin{aligned}&\color {blue}a_{k+1}=(2k+1)^{2}\end{aligned}}}
a k + 1 = a k + 8 k a k + 1 = a k ⏟ ( 2 k − 1 ) 2 + 8 k a k + 1 = ( 2 k − 1 ) 2 + 8 k a k + 1 = 4 k 2 − 4 k + 1 + 8 k a k + 1 = 4 k 2 + 4 k + 1 a k + 1 = ( 2 k + 1 ) 2 {\displaystyle {\begin{aligned}&\color {green}a_{k+1}=a_{k}+8k\\&a_{k+1}=\underbrace {a_{k}} _{(2k-1)^{2}}+8k\\&a_{k+1}=(2k-1)^{2}+8k\\&a_{k+1}=4k^{2}-4k+1+8k\\&a_{k+1}=4k^{2}+4k+1\\&a_{k+1}=(2k+1)^{2}\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
