Question

A=1/4+1/4²+1/4³+.....+1/4⁹⁹

a) Rut gon A

b) So sanh A voi 1/3

Answer 1

4A=1+1/4+1/4²+......+1/4⁹⁸

4A - A = ( 1+1/4+1/4²+..........+1/4⁹⁸) - ( 1/4+1/4²+1/4³+.......+1/4⁹⁹)

3A= 1-1/4⁹⁹

A= 1/3 - 1/4⁹⁹ : 3

Mà 1/4⁹⁹ : 3 > 0 => 1/3 - 1/4⁹⁹ : 3 < 1/3

                          Hay A < 1/3

 

Answer 2

a/ Rút gọn:

\(A=\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+....+\frac{1}{4^{99}}.\)

=> \(4A=1+\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+....+\frac{1}{4^{98}}\)

=> \(4A=1+\left(\frac{1}{4}+\frac{1}{4^2}+\frac{1}{4^3}+....+\frac{1}{4^{98}}+\frac{1}{4^{99}}\right)-\frac{1}{4^{99}}\)

<=> \(4A=1+A-\frac{1}{4^{99}}\)

=> \(3A=1-\frac{1}{4^{99}}\)

=> \(A=\frac{1}{3}-\frac{1}{3.4^{99}}\)

b/ Ta có: \(A=\frac{1}{3}-\frac{1}{3.4^{99}}< \frac{1}{3}\)

Answer 3

Ta có:

A=1/4+1/4²+…+1/4⁹⁹

4A=4.(1/4+1/4²+…+1/4⁹⁹)

4A=1+1/4+…+1/4⁹⁸

4A-A=(1+1/4+…+1/4⁹⁸)- (1/4+1/4²+…+1/4⁹⁹)

3A=1-1/4⁹⁹

A=1/3-1/(4⁹⁹.3)

b)Vì A=1/3-1/(4⁹⁹​.3) nên A<1/3 (do 1/4⁹⁹​.3>0)