2條 linear algebra

1 S_1 = 1, S_2 = 5, S_3 = 1 + 4 + 18 = 23, S_4 = 23 + 96 = 119

It seems that S_n = (n + 1)! - 1

Let P(n) be the statement: "S_n = (n + 1)! - 1"

when n = 1, S_1 = 1 = 2! - 1

Assume that P(k) is true: S_k = (k + 1)! - 1

when n = k + 1

S_k + 1

= S_k + (k + 1)(k + 1)!

= (k + 1)! - 1 + (k + 1)(k + 1)!

= (k + 1)![ k + 1 + 1] - 1

= (k + 2)! - 1

So, P(k + 1) is true. By, M.I. for all positive integer n, P(n) is true.

2 A^2 = [1,2,5 ; 0,1,2; 0,0,1]

A^3 = [1,3,9 ; 0,1,3; 0,0,1]

A^4 = [1,4,14 ; 0,1,4; 0,0,1]

It seems that A^n = [1,n,n(n+3)/2 ; 0,1,n; 0,0,1]

Let P(n) be the statement:

A^n = [1,n,n(n+3)/2 ; 0,1,n; 0,0,1]

when n = 1, A^1 = [1,1,(1)(1+3)/2 ; 0,1,1; 0,0,1]

P(1) is true.

Assume that P(k) is true.

A^k = [1,k,k(k+3)/2 ; 0,1,k; 0,0,1]

A^(k+1) = [1,1,2 + k + k(k+3)/2; 0,1,k+1;0,0,1]

= [1,1,(k^2+5k+4)/2; 0,1,k+1;0,0,1]

= [1,1,(k+1)(k+4)/2; 0,1,k+1;0,0,1]

So, P(k+1) is true. By, M.I. for all positive integer n, P(n) is true.