引用第28樓yhmjack1於2009-12-13 14:38發表的 :
有條M.I唔識....
If prove that
1x2x2+2x3x4+....+n(n+1)(2n)=n(n+1)(n+2)(3n+1)
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6
.......
Let P(n) be the proposition 1x2x2+2x3x4+....+n(n+1)(2n) = n(n+1)(n+2)(3n+1)/6
when n = 1
L.H.S = 4
R.H.S = 1(1+1)(1+2)(3+1)/6 = 4
as L.H.S = R.H.S
P(1) is true
Assume P(k) is true for some positive integers k
1x2x2+2x3x4+....+k(k+1)(2k) = k(k+1)(k+2)(3k+1)/6
when n = k+1
L.H.S =1x2x2+2x3x4+....+k(k+1)(2k) + (k+1)(k+2)(2k+2)
= k(k+1)(k+2)(3k+1)/6 + (k+1)(k+2)(2k+2)
=(k+1)(k+2)[k(3k+1) + 12k+12]/6
=(k+1)(k+2)(3k
2 + 13k + 12)/6
=(k+1)(k+2)(3k+4 )(k+3 )/6
R.H.S=(k+1)(k+2)(k+3)(3k+4)/6
so L.H.S = R.H.S
By the principle of MI...................................
8x16x18+9x18x20+....+18x36x38
=2[8x9x16 + 9x10x18 +....18x19x36]
=2[1x2x2+2x3x4+....+18x19x36 - 1x2x2 +.....7x8x14]
=2 * [18(19)(20)(55)/6 - 7(8)(9)(25)/6]
=