\(A=3+3^2+3^3+...+3^{100}\)

\(\Rightarrow3A=3^2+3^3+3^4+...+3^{101}\)

\(3A-A=3^{101}+3^{100}+3^{99}+...+3^2-3^{100}-3^{99}-...-3\)

\(2A=3^{101}-3\)

Ta thấy \(3^{101}-3< 3^{101}-1\)hay 2A<B=>A< B.

\(A=1+3^2+3^4+...+3^{100}\)

\(9A=3^2+3^4+3^6+...+3^{102}\)

\(8A=3^{102}-1\)

\(\Rightarrow8A-26=3^{102}-1-26=3^{102}-27\)

Vì \(3^{102}-27⋮3\)(1)

\(3^{102}-27⋮2\)(\(3^{102}-27\)là số chẵn )      (2)

\(3^{102}-27=9\left(3^{100}-3\right)\)\(\Rightarrow3^{102}-27⋮9\)(3)

Từ (1) , (2), (3) \(\Rightarrow8A-26⋮54\)\(\left(\left(2,3,9\right)=1\right)\)

vậy ...

\(A=1+3^2+3^4+...+3^{100}\)

\(\Leftrightarrow3^2A=3^2\left(1+3^2+3^4+....+3^{100}\right)\)

\(\Leftrightarrow9A=3^2+3^4+3^6+...+3^{102}\)

\(\Leftrightarrow9A-A=\left(3^2+3^4+3^6+....+3^{102}\right)-\left(1+3^2+3^4+...+3^{100}\right)\)

\(\Leftrightarrow8A=3^{102}-1\)

\(\Leftrightarrow8A-26=3^{102}-1-26=3^{102}-27\)

Ta có: \(3^{102}⋮3;27⋮3\Rightarrow3^{102}-27⋮3\left(1\right)\)

\(3^{102}-27⋮2\left(2\right)\)(3^102 -27 là số lẻ)

\(3^{102}-27=\left(3^2\right)^{51}-27=9^{51}-27⋮9\left(3\right)\)

(1)(2)(3) => 8A-26 chia hết cho 54 (đpcm)

Như các bạn đã trình bày: Chúng ta chứng minh được:

\(8A-26=3^{102}-27\)

Ta có: \(3^{102}-27⋮2\)( vì \(3^{102};27\)là số lẻ; hiệu 2 số lẻ là số chẵn )

và \(3^{102}-27=27\left(3^{99}-1\right)⋮27\)

vì ( 27; 2) = 1 và 27.2 = 54 nên: \(3^{102}-27⋮54\)

Ta có:

3A-A=(3²+3³+3⁴+3⁵+...+3¹⁰⁰+3¹⁰¹)-(3+3²+3³+3⁴+...+3¹⁰⁰)=3¹⁰¹-3 =>2A=3¹⁰¹-3 < 3¹⁰¹-3=B

=> A<B (Chứ ko phải A>B)

\(A=\dfrac{4}{3}+\dfrac{10}{9}+\dfrac{28}{27}+....+\dfrac{\left(3^{99}+1\right)}{3^{99}}\)

\(A=\dfrac{4}{3}+\dfrac{10}{3^2}+\dfrac{28}{3^3}+...+\dfrac{\left(3^{99}+1\right)}{3^{99}}\)

\(A=\left(1+\dfrac{1}{3}\right)+\left(1+\dfrac{1}{3^2}\right)+\left(1+\dfrac{1}{3^3}\right)+...+\left(1+\dfrac{1}{3^{99}}\right)\)

\(A=\left(1+1+....+1\right)+\left(\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{99}}\right)\)

\(A=99+\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\right)\)

Gọi \(\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\right)\)là T

\(T=\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\right)\)

\(3T=1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{98}}\)

\(3T-T=\left(1+\dfrac{1}{3}+\dfrac{1}{3^2}+...+\dfrac{1}{3^{98}}\right)-\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\right)\)

\(2T=1-\dfrac{1}{3^{99}}\)

\(T=\left(1-\dfrac{1}{3^{99}}\right):2\)

\(T=\dfrac{1}{2}-\dfrac{1}{3^{99}\cdot2}\)

\(=>A=99+T=99+\dfrac{1}{2}-\dfrac{1}{3^{99}\cdot2}=99,5-\dfrac{1}{3^{99}\cdot2}< 100\)

Vậy A < 100

cảm ơn bn

\(a.A=\frac{1}{2}+\left(\frac{1}{2}\right)^2+\left(\frac{1}{2}\right)^3+...+\left(\frac{1}{2}\right)^{99}\) 

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

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

\(2A-A=1-\frac{1}{2^{99}}\)

\(A=1-\frac{1}{2^{99}}< 1\)

\(b.B=\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+...+\frac{100}{3^{100}}\)

\(3A=1+\frac{2}{3}+\frac{3}{3^2}+...+\frac{100}{3^{99}}\)

\(3A-A=\left(1+\frac{2}{3}+\frac{3}{3^2}+...+\frac{100}{3^{99}}\right)-\left(\frac{1}{3}+\frac{2}{3^2}+\frac{3}{3^3}+...+\frac{100}{3^{100}}\right)\)

\(2A=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{99}}-\frac{100}{3^{100}}\)

\(6A=3+1+\frac{1}{3}+...+\frac{1}{3^{98}}-\frac{100}{3^{99}}\)

\(6A-2A=\left(3+1+\frac{1}{3}+...+\frac{1}{3^{98}}-\frac{100}{3^{99}}\right)-\left(1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{99}}-\frac{100}{3^{100}}\right)\)

\(4A=3-\frac{100}{3^{99}}-\frac{1}{3^{99}}+\frac{100}{3^{100}}\)

\(4A=3-\frac{300}{3^{100}}-\frac{3}{3^{100}}+\frac{100}{3^{100}}\)

\(4A=3-\frac{303}{3^{100}}+\frac{100}{3^{100}}\)

\(4A=3-\frac{203}{3^{100}}< 3\)

\(A< \frac{3}{4}\)

Ủng hộ mk nha ^_^

Bài 1 : \(A=1+3+3^2+...+3^{31}\)

a. \(A=\left(1+3+3^2\right)+...+3^9.\left(1.3.3^2\right)\)

\(\Rightarrow A=13+3^9.13\)

\(\Rightarrow A=13.\left(1+...+3^9\right)\)

\(\Rightarrow A⋮13\)

b. \(A=\left(1+3+3^2+3^3\right)+...+3^8.\left(1+3+3^2+3^3\right)\)

\(\Rightarrow A=40+...+3^8.40\)

\(\Rightarrow A=40.\left(1+...+3^8\right)\)

\(\Rightarrow A⋮40\)

Bài 2:

Ta có: \(C=3+3^2+3^4+...+3^{100}\)

\(\Rightarrow C=(3+3^2+3^3+3^4)+...+(3^{97}+3^{98}+3^{99}+3^{100})\)

\(\Rightarrow3.(1+3+3^2+3^3)+...+3^{97}.(1+3+3^2+3^3)\)

\(\Rightarrow3.40+...+3^{97}.40\)

Vì tất cả các số hạng của biểu thức C đều chia hết cho 40

\(\Rightarrow C⋮40\)

Vậy \(C⋮40\)

Bài 1 : Ta có : S = 1 + 2 + 2² + 2³ + ... + 2⁹

                     2S = 2(1 + 2 + 2² + 2³ + ... + 2⁹)

                     2S = 2 + 2² + 2³ + ... + 2¹⁰

                 2S -  S = (2 + 2² + 2³ + ... + 2¹⁰) - (1 + 2 + 2² + 2³ + ... + 2⁹)

                        S = 2¹⁰ - 1 = 2⁸.4 - 1

Vậy S < 5 x 2⁸

Bn có thể giải cho mik bài2 và bài4 đc ko ngay bây giờ nhé