F4 A maths M.I.

Let P(n)be the proposition,
1/1．2．3+1/2．3．4+....+1/n(n+1)(n+2)=n(n+3)/4(n+1)(n+2)
when n=1,
L.H.S=1/(1．2．3)=1/6
R.H.S=1(1+3)/4(2)(3)=4/24=1/6
L.H.S=R.H.S
So,P(1)is true.
Assume that P(k)is true,
i.e. 1/1．2．3+1/2．3．4+....+1/k(k+1)(k+2)=k(k+3)/4(k+1)(k+2)
when n=k+1,
L.H.S=1/1．2．3+1/2．3．4+....+1/k(k+1)(k+2)+1/(k+1)(k+2)(k+3)
　　=k(k+3)/4(k+1)(k+2)+1/(k+1)(k+2)(k+3)
=[k(k+3)^2+4]/4(k+1)(k+2)(k+3)
=(k+1)(k+1)(k+4)/4(k+1)(k+2)(k+3)
=(k+1)(k+4)/4(k+2)(k+3)
R.H.S=(k+1)(k+3+1)/4(k+1+1)(k+2+1)
=(k+1)(k+4)/4(k+2)(k+3)
L.H.S=R.H.S
So, P(k+1)is true.
By the principle of M.I.,P(n) is true for all integers n.

Let S(n) be the statement 1 / 1．2．3 + 1 / 2．3．4 +......+ 1 / n (n+1) ( n+2) = n(n+3) / 4( n+1 ) ( n+2) is true for all positive integers n.
Consider S(1)
L.H.S= 1/1*2*3=1/6
R.H.S.=1(1+3)/4(1+1)(1+2)=1/6=L.H.S.
Therefore S(1) is true
Assume S(k) is true for some positive integers k
i.e. 1 / 1．2．3 + 1 / 2．3．4 +......+ 1 / k (k+1) ( k+2) = k(k+3) / 4( k+1 ) ( k+2)
Consider S(k+1)
L.H.S
=1 / 1．2．3 + 1 / 2．3．4 +......+ 1 / k (k+1) ( k+2)+1/(k+1)(k+1+1)(k+1+2)
=k(k+3) / 4( k+1 ) ( k+2)+1/(k+1)(k+1+1)(k+1+2)
=k(k+3) / 4( k+1 ) ( k+2)+1/(k+1)(k+2)(k+3)
=(k(k+3)(k+3)+4)/4(k+1)(k+2)(k+3)
=(k^3+6(k^2)+9k+4)/4(k+1)(k+2)(k+3)
(At this rate, factorise the numerator by factor theorem. You should know that in mathematics. and there must be two (k+1))
=(k+1)(k^2+5k+4)/4(k+1)(k+2)(k+3)
=(k+1)(k+1)(k+4)/4(k+1)(k+2)(k+3)
=(k+4)/4(k+2)(k+3)
=(k+1+3)/4(k+1+1)(k+1+2)
=R.H.S.
Therefore S(k+1) is ture
By the principle of mathematical induction, S9n) is true