a-maths

1
let P(n) 1x1!+2x2!+....+nxn!= (n+1)!-1

when n = 1：

1x1! = 1
(1+1)!-1 = 2-1 = 1

P(1) is true

Assume P(k) is true，that is： 1x1!+2x2!+....+kxk!= (k+1)!-1

when n = k+1,

1x1!+2x2!+....+kxk!+ (k+1)x(k+1)!
= (1x1!+2x2!+....+kxk! ) + (k+1)x(k+1)!
= (k+1)!-1 + (k+1)x(k+1)!
= (k+1)!(1+k+1)-1
= (k+1)!(k+2)-1
= (k+2)!-1

P(k+1) is true

By mathematical induction ，for all positive integers n
1x1!+2x2!+....+nxn!= (n-1)!-1
2
p= (x^2+2x+8) / (x-2)

p(x - 2) = x^2 + 2x + 8
px - 2p = x^2 + 2x + 8
x^2 + (2 - p)x + (8 + 2p) = 0

since x is real, b^2 - 4ac > = 0

(2 - p)^2 - 4(1)(8 + 2p) > = 0

(p^2 - 4p + 4) - 32 - 8p > = 0
p^2 - 12p - 28 > = 0
(p - 14) (p + 2) > = 0
p > = 14 or p < = -2
so,
(x^2+2x+8)/(x-2) > = 14 or (x^2+2x+8)/(x-2) < = -2
for l (x^2 +2x+8)/x-2 l
since (x^2+2x+8)/(x-2) > = 14 or (x^2+2x+8)/(x-2) < = -2
the value of l (x^2 +2x+8)/(x-2) l should be greater than or equal to 2