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Tính các tổng sau:
1, S=1-2+3_4+..+25-26
S =-1+3-5+7-...-53+55 ( có 28 số hạng )
= (-1+3)+(-5+7)+...+(-53+55) ( có 28:2=14 nhóm )
= 2+2+...+2
= 2 . 14
= 28
- 135 + 360 + 65 + 40
=(360 + 40) + ( 135 + 65)
= 400 + 200
= 600
2. 463 + 318 + 137 + 22
= (463 + 137) + ( 318 + 22)
= 600 + 340
=940
3. 20 + 21 +22 + ... + 29 + 30
=(20+ 30) + ( 21+ 29) + ( 22+ 28) + (23+ 27) + (24 +26) +25
= 50 + 50 + 50 + 50 + 50 + 25
= 275
a) Đặt: \(A=1+2^2+2^3+...+2^{10}\)
\(\Rightarrow2A=2\left(1+2^2+2^3+...+2^9+2^{10}\right)\)
\(\Rightarrow2A=2+2^3+2^4+...+2^{10}+2^{11}\)
\(\Rightarrow2A-A=\left(2+2^3+2^4+...+2^{10}+2^{11}\right)-\left(1+2^2+2^3+...+2^{10}\right)\)
\(\Rightarrow A=\left(2^3-2^3\right)+\left(2^4-2^4\right)+...+\left(2-1\right)+\left(2^{11}-2^2\right)\)
\(\Rightarrow A=0+0+...+1+\left(2^{11}-2^2\right)\)
\(\Rightarrow A=1+2^{11}-2^2=1+2048-4=2045\)
Vậy: \(1+2^2+2^3+...+2^{10}=2045\)
b)
a] \(60-3\left(x-1\right)=2^3\cdot3\)
\(\Rightarrow60-3\left(x-1\right)=24\)
\(\Rightarrow3\left(x-1\right)=36\)
\(\Rightarrow x-1=12\)
\(\Rightarrow x=13\)
b] \(\left(3x-2\right)^3=2\cdot2^5\)
\(\Rightarrow\left(3x-2\right)^3=2^6\)
\(\Rightarrow\left(3x-2\right)^3=\left(2^2\right)^3\)
\(\Rightarrow3x-2=2^2\)
\(\Rightarrow3x=6\)
\(x=2\)
c] \(5^{x+1}-5^x=500\)
\(\Rightarrow5^x\left(5-1\right)=500\)
\(\Rightarrow5^x\cdot4=500\)
\(\Rightarrow5^x=125\)
\(\Rightarrow5^x=5^3\)
\(\Rightarrow x=3\)
d] \(x^2=x^4\)
\(\Rightarrow x=x^2\)
\(\Rightarrow x-x^2=0\)
\(\Rightarrow x\left(1-x\right)=0\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\1-x=0\end{matrix}\right.\)
\(\Rightarrow\left[{}\begin{matrix}x=0\\x=1\end{matrix}\right.\)
Vì: \(\frac{3}{21}=\frac{3}{21}\)
\(\frac{3}{22}\) < \(\frac{3}{21}\)
\(\frac{3}{23}\) < \(\frac{3}{21}\)
\(\frac{3}{24}\)<\(\frac{3}{21}\)
\(\frac{3}{25}\)< \(\frac{3}{21}\)
.....
\(\frac{2}{29}\)<\(\frac{3}{21}\)
\(\frac{2}{30}\)<\(\frac{3}{21}\)
Nên \(\frac{3}{21}+\frac{3}{22}+\frac{3}{23}+\frac{3}{24}+\frac{3}{25}+...+\frac{3}{29}+\frac{3}{30}\) < \(\frac{3}{21}.10\)
Ta có: \(\frac{3}{21}.10\) = \(\frac{10}{7}\)
Mà \(\frac{10}{7}\) < \(\frac{3}{2}\)
=>\(\frac{3}{21}+\frac{3}{22}+\frac{3}{23}+\frac{3}{24}+\frac{3}{25}+...+\frac{3}{29}+\frac{3}{30}\) < \(\frac{3}{2}\)
Vậy E < M
a)29 . (19 – 13) – 19 . (29 – 13)
= 29 . 6 – 19 . 16
= 174 – 304
= –130.
b)31.(-18)+31.(-81)-31
= 31. [-18 + (-81) - 1 ]
= 31. (-100)
= -3100
c)(7.3-3):(-6)
(7.3-3):(-6)
= (21-3):(-6)
= 18 : (-6)
= 3
d)72:[(-6).2+4)]
= 72 : ( -12 + 4 )
= 72 : -8
= -9
ta thấy \(\frac{1}{20}\)<\(\frac{1}{3}\)
thì \(\frac{1}{20}\)+...+\(\frac{1}{29}\)<\(\frac{1}{20}\)+...+\(\frac{1}{20}\)<\(\frac{1}{3}\)
vậy \(\frac{1}{20}\)+...+\(\frac{1}{29}\)<\(\frac{1}{3}\)
\(\frac{9}{8}-\frac{-11}{8}=\frac{9-\left(-11\right)}{8}=\frac{9+11}{8}=\frac{20}{8}=\frac{5}{2}\)
\(-\frac{1}{2}-\frac{5}{2}=\frac{-1-5}{2}=\frac{-6}{2}=-3\)
\(\frac{4}{3}-\frac{7}{3}=\frac{4-7}{3}=\frac{11}{3}\)
\(\frac{3}{-5}-\frac{-8}{10}=\frac{6}{-10}-\frac{-8}{10}=\frac{-6}{10}-\frac{-8}{10}=\frac{-6-\left(-8\right)}{10}=\frac{-6+8}{10}=\frac{2}{10}=\frac{1}{5}\)
\(\frac{5}{7}-\frac{-3}{21}=\frac{5}{7}-\frac{-1}{7}=\frac{5-\left(-1\right)}{7}=\frac{5+1}{7}=\frac{6}{7}\)
\(\frac{4}{-3}-\frac{9}{27}=\frac{4}{-3}-\frac{1}{3}=\frac{-4}{3}-\frac{1}{3}=\frac{-4-1}{3}=\frac{-5}{3}\)
\(\frac{7}{29}-\frac{9}{29}=\frac{7-9}{29}=\frac{2}{29}\)
\(\frac{-7}{22}-\frac{9}{22}=\frac{-7-9}{22}=\frac{-16}{22}=\frac{-8}{11}\)
\(\frac{-23}{7}-\frac{31}{7}=\frac{-23-31}{7}=\frac{-54}{7}\)
a ) ( -1/6 + 5/12 ) + 7/12
= -1/6 + ( 5/12 + 7/12 )
= -1/6 + 12/12
= -2/12 + 12/12
= -10/12
= -5/6
Bài 2: Tính nhanh
a) 29 + 132 + 237 + 868 + 3
= 29+(132+858)+(237+3)
=29+990+240
=1259