F.4 AM

1)
let S(n) be the statement " 1^3 +3^3 +5^3+...+(2n+1)^3=(n^2)[(2n^2)-1)] "
when n = 1,
LHS = 1^3+3^3 = 28
RHS = 1(2-1) = 1
.'.S(1) is not true.
by mathematical induction,S(n) is not true for all n

2)
(1)
1-(1/2^2)
=3/4
(2)
(1/2^2)-(1/3^2)
= 5/36
(3)
(1/3^2)-(1/4^2)
= 7/144
(4)
(1/n^2)-[1/(n+1)^2]
= 2n+1/n^2(n+1)^2

b,c)
let S(n) be the statement "(3/1^2*2^2)+(5/2^2*3^2)+(7/3^2*4^2)+......+[2n+1/n^2*(n+1)^2]
= (n^2 +2n)/(n+1)^2"
when n =1 ,
LHS = 3/ 1^2*2^2 = 3/4
RHS = 3/4
S(1) is true .
assume S(k) is true , i.e.(3/1^2*2^2)+(5/2^2*3^2)+(7/3^2*4^2)+......+[2k+1/k^2*(k+1)^2]
= (k^2 +2k)/(k+1)^2
when n = k+1 ,
LHS = (3/1^2*2^2)+(5/2^2*3^2)+(7/3^2*4^2)+......+[2k+1/k^2*(k+1)^2]+[2k+3/(k+1)^2*(k+2)^2]
= (k^2 +2k)/(k+1)^2 +[2k+3/(k+1)^2*(k+2)^2]
=[k(k+2)^3 +2k+3]/(k+1)^2(k+2)^2
= [k^4 +6k^3 + 12k^2 +8k +2k+3]/(k+1)^2(k+2)^2
=[k^4 +6k^3 + 12k^2 +10k+3]/(k+1)^2(k+2)^2
= (k+1)^2(k^2+4k+3)/(k+1)^2(k+2)^2
=(k^2+4k+3)/(k+2)^2
=RHS
by mathematical induction, S(n) is true for all natural no. n

2007-09-06 22:12:29 補充：
第一個錯:(1/3^2)-(1-1/4^2)應改為(1/3^2)-(1/4^2)第二個錯:(3/1^2*2^2) (5/2^2*3^2) (7/3^2*4^2) ...... [2n-1/n^2*(n 1)^2]應改為(3/1^2*2^2) (5/2^2*3^2) (7/3^2*4^2) ...... [2n 1/n^2*(n 1)^2]小朱朱打錯lu

2007-09-10 20:19:42 補充：
其實係睇規律ge 啫1-(1/2^2) = (3/1^2*2^2)(1/2^2)-(1/3^2) = (5/2^2*3^2).............................等等(3/1^2*2^2) (5/2^2*3^2) (7/3^2*4^2) ...... [2n-1/n^2*(n 1)^2] = 1-(1/2^2) (1/2^2)-(1/3^2) (1/3^2)-..... (1/n^2)-[1/(n 1)^2] =1 - [1/(n 1)^2] = n(n 2)/(n 1)^2明白嗎?

1 s(n) is not true for all natural no. n
2a 1 3/4
2 5/36
3 7/144
4 2n+1/n^2 (n+1)^2
b (n+1)^2 -1 / (n+1)^2