\(\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)...\left(1+\frac{1}{100}\right)\)

\(=\left(\frac{2}{2}+\frac{1}{2}\right)\left(\frac{3}{3}+\frac{1}{3}\right)...\left(\frac{100}{100}+\frac{1}{100}\right)\)

\(=\frac{3}{2}.\frac{4}{3}...\frac{101}{100}\)

\(=\frac{3.4...101}{2.3...100}\)

\(=\frac{101}{2}\)

\(\left(1+\frac{1}{2}\right)\left(1+\frac{1}{3}\right)...\left(1+\frac{1}{100}\right)\)

\(=\left(\frac{2}{2}+\frac{1}{2}\right)\left(\frac{3}{3}+\frac{1}{3}\right)...\left(\frac{100}{100}+\frac{1}{100}\right)\)

\(=\frac{3}{2}\cdot\frac{4}{3}\cdot...\cdot\frac{101}{100}\)

\(=\frac{101}{2}\)

nếu không biết thì có đc trl không ??? :) 

 Xin tự tloi

\(\frac{7}{4}+3-\frac{1}{2}\)

\(=\frac{17}{4}\)

Học good

\(\frac{17}{4}\)

1+1x0

=1+0

=

\(1+1\cdot0=1\)

Chúc bạn thi tốt

xét vế trái

ta có:đề\(=\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{14}-\frac{1}{17}\) 

\(=\frac{1}{2}-\frac{1}{17}< < \frac{1}{2}\)

vậy vế trái bé hơn \(\frac{1}{2}\)

P/S:  \(< < \)là luôn luôn bé hơn nha

k mình nha bạn

Thiengl2015#

Ta có :

\(\frac{3}{2.5}+\frac{3}{5.8}+\frac{3}{8.11}+...+\frac{3}{14.17}\)

\(=\frac{1}{2}-\frac{1}{5}+\frac{1}{5}-\frac{1}{8}+\frac{1}{8}-\frac{1}{11}+...+\frac{1}{14}-\frac{1}{17}\)

\(=\frac{1}{2}-\frac{1}{17}\)

Mà \(\frac{1}{2}-\frac{1}{17}< \frac{1}{2}\)

Nên \(\frac{3}{2.5}+\frac{3}{5.8}+\frac{3}{8.11}+...+\frac{3}{14.17}< \frac{1}{2}\left(đpcm\right)\)

Đầu tiên,ta chứng minh BĐT phụ \(\frac{\left(x+y\right)^2}{2}\ge2xy\Leftrightarrow\frac{\left(x+y\right)^2-4xy}{2}\ge0\)

\(\Leftrightarrow\left(x-y\right)^2\ge0\) (luôn đúng).Dấu "=" xảy ra khi x = y.

Và BĐT \(\frac{1}{x}+\frac{1}{y}\ge\frac{4}{x+y}\).Áp dụng BĐT AM-GM(Cô si),ta có; \(\frac{1}{x}+\frac{1}{y}\ge\frac{2}{\sqrt{xy}}\ge\frac{2}{\frac{\left(x+y\right)}{2}}=\frac{4}{x+y}\)

Dấu "=" xảy ra khi x = y

\(P=\frac{1}{a^2+b^2}+\frac{1}{2ab}+\frac{1}{2ab}\)\(\ge\frac{4}{a^2+b^2+2ab}+\frac{1}{2ab}\)

\(\ge\frac{4}{\left(a+b\right)^2}+\frac{1}{\frac{\left(a+b\right)^2}{2}}\ge4+\frac{1}{\frac{1}{2}}=6\)

Dấu "=" xảy ra khi a = b và a + b = 1 tức là a=b=1/2

Vậy Min P = 6 khi a = b = 1/2 

\(\frac{x}{24}=\frac{-2}{3}\Leftrightarrow x=\frac{-2\times24}{3}=-16\)

\(\frac{y}{-18}=\frac{-2}{3}\Leftrightarrow y=\frac{-2\times-18}{3}=12\)

\(\frac{-28}{z}=\frac{-2}{3}\Leftrightarrow z=\frac{-28\times3}{-2}=42\)

\(\frac{-10}{t}=\frac{-2}{3}\Leftrightarrow t=\frac{-10\times3}{-2}=15\)

\(\frac{x}{24}=\frac{y}{-18}=\frac{-28}{z}=\frac{-10}{t}=\frac{-2}{3}\)

\(\Rightarrow t=-15\)

\(\Rightarrow z=42\)

\(\Rightarrow y=12\)

\(\Rightarrow x=-16\)

\(S=3+\frac{3}{2}+\frac{3}{2^2}+\frac{3}{2^3}+...+\frac{3}{2^9}\)

\(\Rightarrow2S=6+2+\frac{3}{2}+\frac{3}{2^2}+...+\frac{3}{2^8}\)

\(\Rightarrow2S-S=6+2+\frac{3}{2}+\frac{3}{2^2}+...+\frac{3}{2^8}-3-\frac{3}{2}-\frac{3}{2^2}-...-\frac{3}{2^9}\)

\(\Rightarrow S=6-\frac{3}{2^9}\)