有兩條數唔識點做，關於數學歸納法ge.. 唔該曬！

1 P(n)：n^4 + 2n^3 - n^2 + 14n = 16 能被8整除

當n = 1,n^4 + 2n^3 - n^2 + 14n = 16 能被8整除

P(1) 成立假設當n = k時﹐k^4 + 2k^3 - k^2 + 14k 能被8整除當n = k + 1時﹐(k + 1)^4 + 2(k + 1)^3 - (k + 1)^2 + 14(k + 1)= k^4 + 4k^3 + 6k^2 + 4k + 1 + 2k^3 + 6k^2 + 6k + 2 - k^2 - 2k - 1 + 14k
+ 14= k^4 + 6k^3 + 11k^2 + 22k + 16 = k^4 + 2k^3 - k^2 + 14k + 4k^3 + 12k^2 + 8k + 16= 8M + 4k^2(k + 3) + 8(k + 2)因為k^2(k + 3)一定是偶數﹐因此8M + 4k^2(k + 3) + 8(k + 2) 一定可被8整除

P(k + 1)成立根據數學歸納法﹐對所有正整數n﹐,n^4 + 2n^3 - n^2 + 14n 能被8整除2 P(n)：1 * 2 + 2 * 3 + ... + n(n + 1) = (1/3)n(n + 1)(n + 2)

當n = 1時﹐L.H.S. = 1 * 2 + 2 * 1 = 4 = (1/3)(2)(2 + 1)(2 + 2) = R.H.S.假設當n = k時﹐1 * 2 + 2 * 3 + ... + k(k + 1) = (1/3)k(k + 1)(k + 2)

當n = k + 1時﹐L.H.S. = 1 * 2 + 2 * 3 + ... + (k + 1)(k + 2)=(1/3)k(k + 1)(k + 2) + (k + 1)(k + 2)= (k + 1)(k + 2)(k/3 + 1)= (1/3)(k + 1)(k + 2)(k + 3)= R.H.S.因此n = k + 1時﹐P(k + 1)成立根據數學歸納法﹐對所有正整數n﹐1 * 2 + 2 * 3 + ... + n(n + 1) = (1/3)n(n + 1)(n + 2)

1 * 3 + 2 * 4 + 3 * 5 + ... + 50* 52= 1 * (2 + 1) + 2 * (3 + 1) + 3 * (4 + 1) + ... + 50 * (51 + 1)= (1 * 2 + 2 * 3 + ... + 50 * 51) + (1 + 2 + 3 + .. + 50)= (1/3)(50)(51)(52) + 50 * 51/2= 45475