Question

Chứng minh rằng P =(1+1/2)(1+1/2^2)....(1+1/2^200) > 3

Answer 1

eh hôm nay mình vừa học dạng này xong; P<3 nhé

P = 3/2 * 2^2+1/2^2 *... * 2^200+1/2^200

Mà 2^2+1/2^2 < 2^2+1-2/2^2-2 = 2^2-1/2^2-2 = 2^2-1/2

2^3+1/2^3 < 2^3+1-2/2^3-2 = 2^3-1/2^3-2 = 2^3-1/2(2^2-1)

...

2^200+1/2^3 < 2^100+1-2/2^100-2 = 2^100-1/2^100-2 = 2^100-1/2(2^199-1)

=> P < 3/2 * 2^2-1/2 * 2^3/2(2^2-1)*...* 2^200-1/2(2^199-1)

=3/2 * 1/2 * 1/2 * 1/2 ...* 1/2 (199 thừa số 1/2) * (2^200-1)

=3/2 * 2^200-1/2^199

= 3 * 2^200-1/2^200

= 3* (1- 1/2^200) < 3*1 = 3

=> đpcm