M=3+32+33+...+3n
=>3M=32+33+34+...+3n+1
=>3M-M=3n+1-3
=>2M=3n+1-3
=>M=\(\frac{3^{n+1}-3}{2}\)
\(N=\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^n}\)
=>3N\(=1+\frac{1}{3}+\frac{1}{3^2}+...+\frac{1}{3^{n-1}}\)
=>3N-N=\(1-\frac{1}{3^n}\)
=>2N=\(1-\frac{1}{3^n}\Rightarrow N=\frac{1-\frac{1}{3^n}}{2}\)
tính hợp lí ( nếu có thể )
\(B=\left(\frac{2}{3}-\frac{1}{4}+\frac{5}{11}\right):\left(\frac{5}{12}+1-\frac{7}{11}\right)\)
\(C=\left(-\frac{14}{33}\right).2\frac{4}{9}+\frac{48}{25}:\frac{27}{25}\)
\(D=\left(3-2\frac{1}{3}+\frac{1}{4}\right):\left(4-5\frac{1}{6}+2\frac{1}{4}\right)\)
\(G=\left(7\frac{1}{9}-2\frac{14}{15}\right):\left(2\frac{2}{3}-6\frac{2}{3}\right)-\frac{32}{45}\)
\(H=-\frac{1}{7}.\left(9\frac{1}{2}-8,75\right):\frac{2}{7}+0,625:1\frac{2}{3}\)
Bài 1: Chứng tỏ các tổng sau không là số tự nhiên:
a. A= \(\frac{1}{2}\)+\(\frac{1}{3}\)+\(\frac{1}{4}\)
b. B= \(\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+...+\frac{1}{8}\)
c. C= \(\frac{3}{10}+\frac{3}{11}+\frac{3}{12}+\frac{3}{13}+\frac{3}{14}\)
Bài 2: Chứng tỏ rằng:
a. A= \(\frac{1}{11}+\frac{1}{12}+\frac{1}{13}+\frac{1}{14}+...+\frac{1}{20}>\frac{1}{2}\)
b. B=\(\frac{1}{50}+\frac{1}{51}+\frac{1}{52}+...+\frac{1}{99}>\frac{1}{2}\)
c. C= \(\frac{1}{10}+\frac{1}{11}+\frac{1}{12}+\frac{1}{100}>1\)
d. D=\(\frac{1}{41}+\frac{1}{42}+\frac{1}{43}+...+\frac{1}{80}>\frac{7}{12}\)
Bài 3: Cho S= \(\frac{1}{31}+\frac{1}{32}+\frac{1}{33}+...+\frac{1}{60}.\)Chứng minh rằng \(\frac{3}{5}< S< \frac{4}{5}\)
Bài 4: Cho B= \(\frac{10n}{5n-3}\), tìm số nguyên n để:
a. B có giá trị nguyên b. B có GTLN
1) Tính:
a) \(\frac{\frac{1}{9}-\frac{5}{6}-4}{\frac{7}{12}-\frac{1}{36}-10}\)
b) \(\frac{\frac{15}{8}-\frac{15}{6}-\frac{15}{32}+\frac{15}{64}}{3-\frac{3}{2}-\frac{3}{4}+\frac{3}{8}}\)
Tính :
\(A=\frac{3}{1}+\frac{3}{1+2}+\frac{3}{1+2+3}+\frac{3}{1+2+3+4}+...+\frac{3}{1+2+3+4+....+100}\)
Tính :
\(\left(\frac{3}{5}-\frac{1}{2}+\frac{7}{3}\right)-\left(\frac{1}{3}-\frac{5}{2}+\frac{1}{5}\right)-\left(-\frac{3}{5}+3-\frac{5}{3}\right)\)
tính nhanh
A=\(\frac{1}{3}+\frac{1}{3^2}+\frac{1}{3^3}+...+\frac{1}{3^8}\)
Tính:
a) A = \(\frac{1}{1.2.3}+\frac{1}{2.3.4}+...+\frac{1}{98.99.100}\)
b) \(B=3+\frac{3}{1+2}+\frac{3}{1+2+3}+...+\frac{3}{1+2+...+100}\)
Tính :
\(A=\frac{2.2016}{1+\frac{1}{1+2}+\frac{1}{1+2+3}+\frac{1}{1+2+3+4}+...+\frac{1}{1+2+3+4+...+2016}}\)
Tính giá trị của biểu thức, tính nhanh nếu có thể
9) \(\frac{13}{50}\) + 9% + \(\frac{41}{100}\) + 24%
10) 2011,2012 . 2000 ( \(\frac{5}{6}\) - \(\frac{1}{3}\) - 50% ) . 1,234 + 10
11) \(\frac{120-0,5.40.5.0,2.20.0,25-20}{1+5+9+........+33+37}\)
12) \(6\frac{2}{7}+7\frac{3}{5}+8\frac{6}{9}+9\frac{1}{4}+\frac{2}{5}+\frac{5}{7}+\frac{1}{3}+\frac{3}{4}+1997\)
