F.4 M.I. 1條 HELP~~~~~~~~~~~~~~

When n = 1 ,LHS = 2^(1-1) = 1RHS = 2^(2-3) + 2^(1-2) = 0.5 + 0.5 = 1Assume that when n = k the statement is true ,i.e.1+2+3+4+.....+2^(k-1) = 2^(2k-3)+2^(k-2) When n = k+1 ,1+2+3+4+.....+2^(k)= 1+2+3+4+...+2^(k-1) + [2^(k-1) + 1] + [2^(k-1) + 2] +...+ [2^(k-1) + 2^(k-1)]= 2^(2k-3)+2^(k-2) + [2^(k-1) + 2^(k-1) + ... + 2^(k-1)] + 1+2+3+4+...+2^(k-1)= 2^(2k-3)+2^(k-2) + [2^(k-1) + 2^(k-1) + ... + 2^(k-1)] + 2^(2k-3)+2^(k-2)..............................{............2 ^ (k-1) terms }..................= 2 * [2^(2k-3) + 2^(k-2)] + 2^(k-1) * 2^(k-1) = 2^(2k-2) + 2^(k-1) + 2^(2k-2)= 2^(2k-1) + 2^(k-1) = 2^(2(k+1) - 3) + 2^((k+1) - 2) (Proved)


2011-04-05 11:08:56 補充：
For example :

1+2+3+4+...+2^4 ........( = 1+2+3+4+...+16)

= (1+2+3+4+...+2³) + (2³+1) + (2³+2) + ... + (2³+2³)

= (1+2+3+4+...+8) + (8+1) + (8+2) + ... + (8+8)

= (1+2+3+4+...+8) + (9) + (10) + ... + (16)

2011-04-05 11:09:03 補充：
呢個是 1 2 3 4 逐項加 1 咁去 , 所以 2^(k-1) 下一項是 2^(k-1) +1 ,
2^(k-1) +2 , .... 一路最尾到 2^(k-1) + 2^(k-1) 恰好 = 2 * (2^(k-1) = 2^k

不知大大貼http://i.imgur.com/yeQV9.gif 這圖 的作用是什麼呢?_?

http://i.imgur.com/yeQV9.gif

令P(n): 對所有正整數n﹐1+2+3+4+...+2^(n-1) = 2^(2n-3) + 2^(n-2)

當n = 1時. L.H.S. = 1 = 1/2 + 1/2 = 1.

P(1) 正確。

假定P(k)是正確的﹐即1+2+3+4+...+2^(k-1) = 2^(2k-3) + 2^(k-2)

當n = k + 1 時3

L.H.S.

= 1+2+3+4+...+2^(k-1) + 2^(k-1) + 1 + ... + 2^k

= 2^(2k-3) + 2^(k-2) + 2^(k-1) + 1 + ... + 2^k

= 2^(2k-3) + 2^(k-2) + [2^(k-1) + 1 + 2^k](2^(k-1))/2

= 2^(2k-3) + 2^(k-2) + [3*2^(k-1) + 1](2^(k-2))

= 2^(2k-3) + 2^(k-1) + 3*2^(2k-3)

= 2^(2k-1) + 2^(k-1)

因此P(k+1)正確。

根據數學歸納法﹐對所有正整數n﹐命題成立。