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Ta có:
A=1+1/3+1/32+1/33+...+1/32014
=>3A=3+1/32+1/33+1/34+...+1/32015
=>2A=2+1/32015-1/3
=>A=1+2/32015-2/3
OK!
a) S= 1+2+22+...+29
2S=2+22+23+...+210
2S-S=(2+22+23+...+210)-(1+2+23+...+29)
S=210-1
5.28=2.2+1.28=1+22.28=1+210
=>S=5.28
b) A=1+2+22+....+2100
2A=2+22+23+...+2101
2A-A=(2+22+23+...+2101)-(1+2+22+...+2100)
A=2101-1
=> A<2101
\(3^{21}=3.3^{20}=3.\left(3^2\right)^{10}=3.9^{10}\)
\(2^{31}=2.2^{30}=2.\left(2^3\right)^{10}=2.8^{10}\)
Thấy: 3 > 2 và 910 > 810
Nên \(3^{21}>2^{31}\)
Bài 2:
\(A=1+2+2^2+.....+2^{100}\)
\(2A=2+2^2+.......+2^{101}\)
\(2A-A=\left(2-2\right)+\left(2^2-2^2\right)+......+2^{101}-1\)
Vậy A = 2101 - 1
\(A=\left[\frac{1}{2^2}-1\right]\left[\frac{1}{3^2}-1\right]\left[\frac{1}{4^2}-1\right]\cdot...\cdot\left[\frac{1}{100^2}-1\right]\)
\(=\frac{-3}{2^2}\cdot\frac{-8}{3^2}\cdot\frac{-15}{4^2}\cdot...\cdot\frac{-9999}{100^2}\)
\(=\frac{-1\cdot3}{2\cdot2}\cdot\frac{-2\cdot4}{3\cdot3}\cdot\frac{-3\cdot5}{4\cdot4}\cdot...\cdot\frac{-99\cdot101}{100\cdot100}\)
\(=\frac{-1\cdot2\cdot3\cdot...\cdot99}{2\cdot3\cdot...\cdot100}\cdot\frac{3\cdot4\cdot5\cdot...\cdot101}{2\cdot3\cdot...\cdot100}\)
\(=-\frac{1}{100}\cdot\frac{101}{2}=-\frac{101}{200}\)
Mà \(-\frac{101}{200}< -\frac{1}{2}\)
nên \(A< -\frac{1}{2}\)
\(A=\left(\frac{1}{2^2}-1\right)\left(\frac{1}{3^2}-1\right)\left(\frac{1}{4^2}-1\right)...\left(\frac{1}{100^2}-1\right)\)
\(A=\left(\frac{1}{4}-1\right)\left(\frac{1}{9}-1\right)\left(\frac{1}{16}-1\right)...\left(\frac{1}{10000}-1\right)\)
\(A=\frac{-3}{4}.\frac{-8}{9}.\frac{-15}{16}...\frac{-9999}{10000}\)
\(A=\frac{-1.3}{2.2}.\frac{-2.4}{3.3}.\frac{-3.5}{4.4}...\frac{-99.101}{100.100}\)
\(A=\frac{\left(-1\right)\left(-2\right)\left(-3\right)...\left(-99\right)}{2.3.4...100}.\frac{3.4.5...101}{2.3.4...100}\)
\(A=-\frac{1}{100}.\frac{101}{2}\)
\(A=-\frac{101}{200}\)
\(\text{Vậy A=}-\frac{101}{200}\)