一條關於數學歸鈉法既問題(急)

1x2x3+2x3x4+3x4x5+.....+n(n+1)(n+2)=n(n+1)(n+2)(n+3)/4

Let S be the statement

When n=1, LHS= 1(1+1)(1+2)
= 1x2x3
RHS= 1(1+1)(1+2)(1+3)/4
= 1x2x3x4/4
=1x2x3
therefore, S is true for n=1

Assume S is true for n=k, where k is an integer
i.e. 1x2x3+2x3x4+3x4x5+.....+k(k+1)(k+2)=k(k+1)(k+2)(k+3)/4

Add (k+1)(k+1+1)(k+1+2) to both sides

1x2x3+2x3x4+.....+k(k+1)(k+2)+ (k+1)(k+1+1)(k+1+2) =k(k+1)(k+2)(k+3)/4 + (k+1)(k+1+1)(k+1+2)
= k(k+1)(k+2)(k+3)/4 + (k+1)(k+2)(k+3)
= k(k+1)(k+2)(k+3)/4 +4(k+1)(k+2)(k+3)/4
= (k+1)(k+2)(k+3)(k +4)/4

therefore, if S is true for n=k, it is also true for n=k+1. By the principle of mathematical induction, S is true for all positive integers.

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alfred_miles兄的答案原全正確。