S = 1 + 3 + 3² + 3³ + 3⁴ + 3⁵ + 3⁶ + 3⁷ + 3⁸ + 3⁹ = (1 + 3) + (3² + 3³) + (3⁴ + 3⁵) + (3⁶ + 3⁷) + (3⁸ + 3⁹) = 1.(1 + 3) + 3².(1 + 3) + 3⁴.(1 + 3) + 3⁶​.(1 + 3) + 3⁸​.(1 + 3) = (1 + 3).(1 + 3² + 3⁴ + 3⁶ + 3⁸) = 4.(1 + 3² + 3⁴ + 3⁶ + 3⁸) => S ⋮ 4. Vậy S ⋮ 4 (đpcm)

gải giúp mình với

a) \(\left(12x-4^3\right).8^3=4.8^4\)

\(12x-4^3=32\)

12x  = 96

x = 8

b) \(\left(3x-2^4\right).7^3=2.7^4\)

3x - 2⁴ = 14

3x = 30

x = 10

Thanks you ! Mình biết làm rồi !

\(B=\left(3+3^3+3^5\right)+3^6\left(3+3^3+3^5\right)+.............+3^{24}\left(3+2^3+3^5\right)\)

\(B=273+273\cdot3^6+.............+273\cdot3^{24}\)

\(B=273\left(1+3^6+.......+3^{24}\right)⋮273\)

\(A=\left(5+5^2\right)+\left(5+5^2\right)5^2+\left(5+5^2\right)5^4+\left(5+5^2\right)5^6+\left(5+5^2\right)5^8\)

\(A=30+30\cdot5^2+30\cdot5^4+30\cdot5^6+30\cdot5^8\)

\(A=30\left(1+5^2+5^4+5^6+5^8\right)⋮30\)

cảm ơn 

1+7+7 mũ 2+7 mũ 3......+7 mũ 100.Tính a,a là tổng dãy số trên 

\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{10^2}\)

\(< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{9.10}\)

\(=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+...+\frac{10-9}{9.10}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{9}-\frac{1}{10}\)

\(=1-\frac{1}{10}< 1\)

\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{10^2}\\ A< \frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{9\times10}\\ A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{9}-\frac{1}{10}=1-\frac{1}{10}\\ A< \frac{9}{10}< 1\Rightarrow A< 1\)

1; 7³.5².5⁴.7⁶:(5⁵.7⁸)

= (7³.7⁶).(5².5⁴) : (5⁵.7⁸)

= 7⁹.5⁶: (5⁵.7⁸)

= (7⁹:7⁸).(5⁶:5⁵)

= 7.5

= 35

2; 3³.a⁷.3.a²:(3⁴.a⁶)

= (3³.3).(a⁷.a²): (3⁴.a⁶)

= 3⁴.a⁹: (3⁴.a⁶)

= (3⁴:3⁴).(a⁹:a⁶)

= a³

3; 7³.11⁴.a⁸.b⁷: 7².11².a⁵.b⁶

=  (7³:7²).(11⁴.11²).(a⁸.a⁵).(b⁷.b⁶)

= 7.11⁶.a¹³.b¹³