Question

Chứng minh rằng:   1/3+2/3^2+3/3^3+....+100/3^100<3/4

Answer 1

 Đặt A = 1/3 + 2/3² + 3/3³ + 4/3^4 + ... + 100/3^100

=> 3A= 1 + 2/3 + 3/3² + 4/3³ + .... + 100/3^99

=> 3A-A = 1 + (2/3 - 1/3) + (3/3² - 2/3²) +...+ (100/3^99 - 99/3^99) - 100/3^100

=> 2A= 1+ 1/3 + 1/3² + 1/3³ +...+ 1/3^99 - 100/3^100

Đặt B = 1/3 + 1/3² + 1/3³ +...+ 1/3^99

=> 3B = 1 + 1/3 + 1/3² + 1/3³ +...+ 1/3^98

=> 2B = 1 - 1/3^99 => B = (1 - 1/3^99)/2

Thay vào 2A => 2A= 1+ 1/2 - 1/(2x3^99) - 100/3^100 < 1+ 1/2 = 3/2

=> A < 3/4

Vậy __________________

Answer 2

Đặt A = 1/3 + 2/3² + 3/3³ + 4/3^4 + ... + 100/3^100

=> 3A= 1 + 2/3 + 3/3² + 4/3³ + .... + 100/3^99

=> 3A-A = 1 + (2/3 - 1/3) + (3/3² - 2/3²) +...+ (100/3^99 - 99/3^99) - 100/3^100

=> 2A= 1+ 1/3 + 1/3² + 1/3³ +...+ 1/3^99 - 100/3^100

Đặt B = 1/3 + 1/3² + 1/3³ +...+ 1/3^99

=> 3B = 1 + 1/3 + 1/3² + 1/3³ +...+ 1/3^98

=> 2B = 1 - 1/3^99 => B = (1 - 1/3^99)/2

Thay vào 2A => 2A= 1+ 1/2 - 1/(2x3^99) - 100/3^100 < 1+ 1/2 = 3/2

=> A < 3/4

Vậy..............

Answer 3

Đặt A = ...........

Nhân 1/3A rồi xét A - 1/3A

Answer 4

A<3/3

Answer 5

co ai biet lam bai nay ko

A=(1/1+1/2+...+1/98)*2*3*...*98 cmr A chia het cho 99

Answer 6

Này bạn:3/2>3/4 nhé.Tính lại đi

Answer 7

mình xin lỗi nhưng mình ko biết làm

Answer 8

to¸n chuyªn ®ª mµ sao biÕt lµm d­îc 

Answer 9

Nguyễn Gia Huy phần đó là 1+1/2=3/2 

Chứ không phải tất cả gộp lại bằng 3/2

Answer 10

A=1/3+2/3^2+...+100/3^100

3A=1+2/3+3/3^2+...+100/3^99

2A=1+1/3+1/3^2+...+1/3^99-100/3^100

đặt B=1/3+1/3^2+1/3^3+...+1/3^99

3B=1+1/3+1/3^2+...+1/3^98

2B=1-1/3^99

B=(1-1/3^99)*1/2=1/2-1/(2*3^99)

suy ra 2A=1+1/2-1/(2*3^99)-100/3^100

A=1/2+1/2^2-1/(2^2*3^99)-100/3^100

=3/4-1/(2^2*3^99)-100/3^100<3/4

suy ra A<3/4