數學A-maths問題?About MI~!

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Σ1/(r(r+2))
=(1/2)Σ[1/r-1/(r+2)]
=(1/2)(1-1/3+1/2-1/4+1/3-1/5+...+1/n-1/(n+2)]
=(1/2)(1+1/2-1/(n+1)-1/(n+2))
=(1/2)(3/2-1/(n+1)-1/(n+2))
例如代n=2
(1/2)(3/2-1/(n+1)-1/(n+2))
=(1/2)(3/2-1/3-1/4)
=11/24
而Σ1/(r(r+2))=1/3+1/8=11/24

2009-01-07 20:28:56 補充：
(b)
when n=1
LHS=1/3
RHS=(1/2)(3/2-1/2-1/3)=1/3

P(1) IS TRUE

Assume that P(k) is true

2009-01-07 20:29:02 補充：
when n=k+1
Σ1/(r(r+2))
=(1/2)(3/2-1/(k+1)-1/(k+2))+1/(k+1)(k+3)
=(1/2)(3/2-(2k+3)/(k+1)(k+2)+2/(k+1)(k+3))
=(1/2)(3/2-[(2k+3)(k+3)-2(k+2)]/(k+1)(k+2)(k+3))
=(1/2)(3/2-(2k^2+7k+5)/(k+1)(k+2)(k+3))
=(1/2)(3/2-(2k+5)/(k+2)(k+3))
=(1/2)(3/2-1/(k+2)-1/(k+3))

So P(k+1) is true