mathematical induction

用戶9285

2009-07-07 4:02 am

1a. prove, by mathematical induction, that all positive integers n,

1^3+2^3+3^3+..............+n^3= n^2 (n+1)^2 /4

b. find the value of 2^3+6^3+10^3+........+38^3

回答 (1)

用戶7639

2009-07-07 4:16 am

✔ 最佳答案

1a.
For n=1,
L.H.S.= 13=1
R.H.S.= 12 (1+1)2 /4
=1
=L.H.S.
∴The statement is true for n=1.
Assume13+23+33+...+k3= k2 (k+1)2 /4,for some positive integer k.
For n=k+1,
L.H.S.
=13+23+33+...+k3+(k+1)3
= k2 (k+1)2 /4+(k+1)3
=(k+1)2[k2+4(k+1)] /4
=(k+1)2(k2+4k+4)] /4
=[(k+1)2(k+2) 2]/4
=R.H.S.
∴The statement is also true for n=k+1 if it is true for n=k.
By the principle of mathematical induction,the statement is true for all positive integer n.

2009-07-06 20:30:49 補充：
b part
http://h.imagehost.org/0066/M_I.jpg

2009-07-06 20:31:50 補充：
b part
http://h.imagehost.org/0066/M_I.jpg

參考:STEVIE-G™

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