= ｢｢nꞒk｣·a˄(n−k)·b˄k Σk=0,n｣
= ｢n!/((n−k)!·k!)·a˄(n−k)·b˄k Σk=0,n｣ᐥ
(a+b)˄1
= ｢1!/((1−k)!·k!)·a˄(1−k)·b˄k Σk=0,1｣
= a+b
(a+b)˄2
= ｢2!/((2−k)!·k!)·a˄(2−k)·b˄k Σk=0,2｣
= a˄2+2·a·b+b˄2
(a+b)˄(n+1)
= (a+b)˄n·(a+b)
= (a+b)˄n·a+(a+b)˄n·b
= ｢n!/((n−k)!·k!)·a˄(n−k+1)·b˄k Σk=0,n｣
+ ｢n!/((n−k)!·k!)·a˄(n−k)·b˄(k+1) Σk=0,n｣
= a˄(n+1)
+ ｢n!/((n−k)!·k!)·a(n−k+1)·b˄k Σk=1,n」
+ ｢n!/((n−k)!·k!)·a˄(n−k)·b˄(k+1) Σk=0,n−1｣
+ b˄(n+1)
= (n+1)!/((n+1−0)!·0!)·a˄(n+1−0)·b˄0
┄
n!/((n−k)!·k!) = (n+1)!/((n+1−k)!·k!)/(n+1)·(n+1−k)
┄
+ ｢(n+1)!/((n+1−k)!·k!)/(n+1)·(n+1−k)·a˄(n+1−k)·b˄k Σk=1,n｣
┄
｢(k+1)/n Σk=0,n」= ｢k/n Σk=1,n+1」
index shift
｢n!/((n−k)!·k!)·a˄(n−k)·b˄(k+1) Σk=0,n−1｣
= ｢(n+1)!/((n+1−(k+1))!·(k+1)!)/(n+1)·(k+1)·a˄(n+1−(k+1))·b˄(k+1) Σk=0,n−1｣
= ｢(n+1)!/((n+1−k)!·k!)/(n+1)·k·a˄(n+1−k)·b˄k Σk=1,n｣
┄
+ ｢(n+1)!/((n+1−k)!·k!)/(n+1)·k·a˄(n+1−k)·b˄k Σk=1,n｣
+ (n+1)!/((n+1−(n+1))!·(n+1)!)·a˄(n+1−(n+1))·b˄(n+1)
= (n+1)!/((n+1−0)!·0!)·a˄(n+1−0)·b˄0
┄
(n+1)!/((n+1−k)!·k!)/(n+1)·(n+1−k) + (n+1)!/((n+1−k)!·k!)/(n+1)·k
= (n+1)!/((n+1−k)!·k!)·(1/(n+1)·(n+1−k) + 1/(n+1)·k)
= (n+1)!/((n+1−k)!·k!)·((n+1−k)/(n+1) + k/(n+1))
= (n+1)!/((n+1−k)!·k!)
┄
+ Σ (n+1)!/((n+1−k)!·k!)·a˄(n+1−k)·b˄k「k=1,n」
+ (n+1)!/((n+1−(n+1))!·(n+1)!)·a˄(n+1−(n+1))·b˄(n+1)
= ｢(n+1)!/((n+1−k)!·k!)·a˄(n+1−k)·b˄k Σk=0,n+1｣
