(3/10:-2/5):(-2/3.4)

=-3/4:(-8/3)

=9/32

_HT_

\(\left(\frac{3}{10}\div\frac{-2}{5}\right)\div\left(\frac{-2}{3}\times4\right)\)

\(=\)\(\left(\frac{3}{10}\times\frac{-5}{2}\right)\div\left(\frac{-2}{3}\times4\right)\)

\(=\)\(\frac{-15}{20}\div\frac{-8}{3}\)

\(=\)\(\frac{-3}{4}\div\frac{-8}{3}\)

\(=\)\(\frac{-3}{4}\times\frac{-3}{8}\)

\(=\)\(\frac{9}{32}\)

Để A đạt giá trị nguyên thì 4n-2 chia hết cho 2n+3

Mà 2(2n+3) chia hết cho 4n-2 hay 4n+6 chia hết cho 2n+3

\(\Rightarrow\)(4n-2)-(4n+6) chia hết cho 2n+3

(4n-4n)-(2+6) chia hết cho 2n+3

-8 chia hết cho 2n+3

\(\Rightarrow2n+3\inƯ\left(-8\right)\)

\(\Rightarrow2n+3\in\left\{1;2;4;8;-1;-2;-4;-8\right\}\)

\(\Rightarrow n\in\left\{-1;-\frac{1}{2};\frac{1}{2};\frac{5}{2};-2;-\frac{5}{2};-\frac{7}{2};-\frac{11}{2}\right\}\)

_HT_

A =2⁰+2+2²+2³+2⁴+....+2⁹⁹

A=(1+2+4+8+16)+(2⁵+2⁶+2⁷+2⁸+2⁹)+....+(2⁹⁵+2⁹⁶+2⁹⁷+2⁹⁸+2⁹⁹)

A=31+2⁵.(1+2+4+8+16)+...+2⁹⁵(1+2+4+8+16)

A=31+2⁵.31+..+2⁹⁵.31

A=31.(1+2⁵+...+2⁹⁵)

\(\Rightarrow A⋮31\)

_HT_

Gọi biểu thức này là \(A\)

Đặt \(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+......+\frac{1}{302^2}\)

Với mọi số tự nhiên ta luôn có : \(\left(n-1\right)^2< n^2< \left(n+1\right)^2\)

\(\Rightarrow\frac{1}{\left(n+1\right)^2}< \frac{1}{n^2}< \frac{1}{\left(n-1\right)^2}\)

Áp dụng ta có :

\(\Rightarrow A< \frac{1}{3^2}+\frac{1}{4^2}+\frac{1}{5^2}+......+\frac{1}{30^2}\)

\(\Rightarrow A< \frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+.......+\frac{1}{29.30}\)

\(\Rightarrow A< \frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+.........+\frac{1}{29}-\frac{1}{30}\)

\(\Rightarrow A< \frac{1}{2}-\frac{1}{30}< \frac{1}{2}< 1\)

\(\Rightarrow A< 1\left(đpcm\right)\)

TA CÓ:

\(\frac{1}{2^2}< \frac{1}{1.2};\frac{1}{3^2}< \frac{1}{2.3};.....;\frac{1}{30^2}< \frac{1}{29.30}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{30^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{29.30}\)

ta lại có:\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{29.30}\)

\(=\frac{2-1}{1.2}+\frac{3-2}{2.3}+\frac{4-3}{3.4}+....+\frac{30-29}{29.30}\)

\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+....+\frac{1}{29}-\frac{1}{30}\)

\(=1-\frac{1}{30}< 1\)

mà \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{30^2}< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{29.30}\)

\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{30^2}< 1-\frac{1}{30}< 1\left(đpcm\right)\)

=4 đúng ko?

=4

HT

K cho mk nhá

\(7^{2022}.7:7^{2020}-75:25+5.2^3-2022^0\)

\(=7^3-3+5.8-1\)

\(=343-3+40-1\)

\(=381\)

7²⁰²² . 7 : 7²⁰²⁰ - 75 : 25 + 5 . 2³ - 2022⁰

= 7²⁰²³ : 7²⁰²⁰ - 75 : 25 + 5 . 8 - 1

= 7³ - 3 + 40 - 1

= 343 - 3 + 40 - 1

= 340 + 40 - 1

= 380 - 1

= 379

HT

HT