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

\(A=-\frac{1}{2}-\frac{3}{5}+\frac{1}{3}=-\frac{23}{30}\)

\(B=-\frac{7}{51}-\frac{4}{13}+\frac{17}{51}+\frac{-9}{13}-\frac{-41}{51}\)

\(=\left(-\frac{7}{51}+\frac{17}{51}+\frac{41}{51}\right)-\left(\frac{4}{13}+\frac{9}{13}\right)\)

\(=1-1=0\)

bằng 2

= 2 nhé

tích mik nha

@tienloc

100 nhân với 8\7

=38