1 ∗ 2 n + 2 ∗ 2 n − 1 + 3 ∗ 2 n − 2 + ⋯ + ( n + 1 ) 2 0 = 2 n + 2 − ( n + 3 ) {\displaystyle 1*2^{n}+2*2^{n-1}+3*2^{n-2}+\cdots +(n+1)2^{0}=2^{n+2}-(n+3)}
L : ( n + 1 ) 2 0 = 2 ∗ 1 = 1 → 2 + 2 ∗ 2 = 4 R : 2 n + 2 − ( n + 3 ) = 2 3 − 4 = 4 4 = 4 {\displaystyle {\begin{aligned}&L:(n+1)2^{0}=2*1=1\rightarrow 2+2*2=4\\&R:2^{n+2}-(n+3)=2^{3}-4=4\\&4=4\\\end{aligned}}}
1 ∗ 2 k + 2 ∗ 2 k − 1 + 3 ∗ 2 k − 2 + ⋯ + ( k + 1 ) 2 0 = 2 k + 2 − ( k + 3 ) {\displaystyle 1*2^{k}+2*2^{k-1}+3*2^{k-2}+\cdots +(k+1)2^{0}=2^{k+2}-(k+3)}
1 ∗ 2 k + 1 + 2 ∗ 2 k + ⋯ + ( k + 1 ) 2 0 + ( k + 1 ) ∗ 2 + ( k + 2 ) ∗ 2 0 = 2 k + 3 − ( k + 4 ) 2 ( 1 ∗ 2 k + 2 ∗ 2 k − 1 + 3 ∗ 2 k − 2 + ⋯ + ( k + 1 ) ∗ 2 0 ⏟ = 2 k + 2 − ( k + 3 ) + ( k + 2 ) 2 0 = 2 k + 3 − ( k + 4 ) 2 [ 2 k + 2 − ( k + 3 ) ] + ( k + 2 ) = 2 k + 3 − ( k + 4 ) 2 k + 3 − 2 k − 6 + k + 2 = 2 k + 3 − k − 4 2 k + 3 − k − 4 = 2 k + 3 − k − 4 0 = 0 {\displaystyle {\begin{aligned}&1*2^{k+1}+2*2^{k}+\cdots +(k+1)2^{0}+(k+1)*2+(k+2)*2^{0}=2^{k+3}-(k+4)\\&2\underbrace {(1*2^{k}+2*2^{k-1}+3*2^{k-2}+\cdots +(k+1)*2^{0}} _{=2^{k+2}-(k+3)}+(k+2)2^{0}=2^{k+3}-(k+4)\\&2[2^{k+2}-(k+3)]+(k+2)=2^{k+3}-(k+4)\\&2^{k+3}-2k-6+k+2=2^{k+3}-k-4\\&2^{k+3}-k-4=2^{k+3}-k-4\\&0=0\\\end{aligned}}}
הטענה נכונה עבור כל n טבעי, ע"פ שלושת שלבי האינדוקציה.
טבעי
![{\displaystyle {\begin{aligned}&1*2^{k+1}+2*2^{k}+\cdots +(k+1)2^{0}+(k+1)*2+(k+2)*2^{0}=2^{k+3}-(k+4)\\&2\underbrace {(1*2^{k}+2*2^{k-1}+3*2^{k-2}+\cdots +(k+1)*2^{0}} _{=2^{k+2}-(k+3)}+(k+2)2^{0}=2^{k+3}-(k+4)\\&2[2^{k+2}-(k+3)]+(k+2)=2^{k+3}-(k+4)\\&2^{k+3}-2k-6+k+2=2^{k+3}-k-4\\&2^{k+3}-k-4=2^{k+3}-k-4\\&0=0\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc305e5b5cf9ff89efa54c9776562d8f846dfbdc)
