a 1 = 1 a n + 1 = a n ∗ 8 n 2 + n a n = 2 n 3 − n {\displaystyle {\begin{aligned}&a_{1}=1\\&a_{n+1}=an*8^{n^{2}+n}\\&a_{n}=2^{n^{3}-n}\\\end{aligned}}}
a n = 2 n 3 − n = 2 1 3 − 1 = 1 a 1 = 1 1 = 1 √ {\displaystyle {\begin{aligned}&a_{n}=2^{n^{3}-n}=2^{1^{3}-1}=1\\&a_{1}=1\\&1=1\surd \\\end{aligned}}}
a k = 2 k 3 − k {\displaystyle \ a_{k}=2^{k^{3}-k}}
a k + 1 = 2 ( k + 1 ) 3 − k − 1 a k + 1 = 2 k 3 + 3 k 2 + 3 k + 1 − k − 1 a k + 1 = 2 k 3 + 3 k 2 + 2 k {\displaystyle {\begin{aligned}&a_{k+1}=2^{(k+1)^{3}-k-1}\\&a_{k+1}=2^{k^{3}+3k^{2}+3k+1-k-1}\\&a_{k+1}=2^{k^{3}+3k^{2}+2k}\\\end{aligned}}}
a k + 1 = a k ⏟ 2 k 3 − k ∗ 8 k 2 + k a k + 1 = 2 k 3 − k ∗ 8 k 2 + k a k + 1 = 2 k 3 − k ∗ 2 3 ( k 2 + k ) a k + 1 = 2 k 3 − k + 3 ( k 2 + k ) a k + 1 = 2 k 3 − k + 3 k 2 + 3 k a k + 1 = 2 k 3 + 3 k 2 + 2 k {\displaystyle {\begin{aligned}&a_{k+1}=\underbrace {a_{k}} _{2^{k^{3}-k}}*8^{k^{2}+k}\\&a_{k+1}=2^{k^{3}-k}*8^{k^{2}+k}\\&a_{k+1}=2^{k^{3}-k}*2^{3(k^{2}+k)}\\&a_{k+1}=2^{k^{3}-k+3(k^{2}+k)}\\&a_{k+1}=2^{k^{3}-k+3k^{2}+3k}\\&a_{k+1}=2^{k^{3}+3k^{2}+2k}\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
