✔ 最佳答案
1x32+2x52+3x72+...+n(2n+1)2=n(n+1)(6n2+14n+7) /6 ...........
Therefore, by principle of mathematical induction, expression is true for all positive integer n
當n=1
LHS = 1*32 = 9
RHS = n(n+1)(6n2+14n+7) /6 = 1(1+1)(6*12+14*1+7)/6 = 9
故當n=1時成立
設 n = k 時成立
則
1x32+2x52+3x72+...+k(2k+1)2=k(k+1)(6k2+14k+7) /6
當n = k + 1
則
LHS = 1x32+2x52+3x72+...+k(2k+1)2+(k+1)(2k+3)2
= k(k+1)(6k2+14k+7) /6 + (k+1)(2k+3)2
= k(k+1)(6k2+14k+7) /6 + 6(k+1)(2k+3)2/6
= (k+1)[k(6k2+14k+7) + 6(2k+3)2]/6
= (k+1)[6k3+14k2+7k + 6(4k2+12k+8)]/6
= (k+1)[6k3+14k2+7k + 24k2+72k+48]/6
= (k+1)[6k3+38k2+79k+48]/6
= (k+1)(k+2)[6k2+26k+27]/6
RHS = (k+1)(K+2)(6(k+1)2+14(k+1)+7) /6
= (k+1)(K+2)(6(k2+2k+1)+14k+14+7) /6
= (k+1)(K+2)(6k2+26k+27) /6
LHS = RHS
所以對任何整數n均成立
hence,or otherwise,find the sum of
11x232+12x252+13x272+...+20x412
1x32 + 2x52 + ……....+20x412
= n(n+1)(6n2+14n+7) /6
= (20)[(20)+1][6(20)2+14(20)+7] /6
= (20)[21][6(400)+280+7] /6
= (20)[21][6(400)+280+7] /6
= 188090
1x32+2x52+3x72+……+10x212
= n(n+1)(6n2+14n+7) /6
= (10)[(10)+1][6(10)2+14(10)+7] /6
= (10)[11][6(100)+140+7] /6
= (10)[11][6(100)+140+7] /6
= 13695
所以
11x232+12x252+13x272+...+20x412
= 188090 – 13695
= 174395
我唔係好明點解
11x232+12x252+13x272+...+20x412=1x32+2x52+3x72+...+20x412-(1x32+2x52+3x72+........+10x212)
因為
1x32+2x52+3x72+…+10x212+11x232+12x252+13x272+...+20x412
將這數列分為兩部份,n由1至10及由11至20兩部份
所以
11x232+12x252+13x272+...+20x412 可以等於由
1x32+2x52+…. +20x412減去前半部1x32+2x52+3x72+…+10x212