3^22 và 2^32
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H9
HT.Phong (9A5)
CTVHS
22 tháng 6 2023
\(A=\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{99}}\)
\(\Rightarrow\dfrac{A}{3}=\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{100}}\)
\(\Rightarrow A-\dfrac{A}{3}=\dfrac{2A}{3}=\left(\dfrac{1}{3}+\dfrac{1}{3^2}+\dfrac{1}{3^3}+...+\dfrac{1}{3^{99}}\right)-\left(\dfrac{1}{3^2}+\dfrac{1}{3^3}+\dfrac{1}{3^4}+...+\dfrac{1}{3^{100}}\right)\)
\(\Rightarrow\dfrac{2A}{3}=\left(\dfrac{1}{3^2}-\dfrac{1}{3^2}\right)+\left(\dfrac{1}{3^3}-\dfrac{1}{3^3}\right)+...+\left(\dfrac{1}{3^{99}}-\dfrac{1}{3^{99}}\right)+\left(\dfrac{1}{3}-\dfrac{1}{3^{100}}\right)=\dfrac{1}{3}-\dfrac{1}{3^{100}}\)
\(\Rightarrow2A=3\cdot\left(\dfrac{1}{3}-\dfrac{1}{3^{100}}\right)\)
\(\Rightarrow\text{A}=\dfrac{1-\dfrac{1}{3^{99}}}{2}\)
\(\Rightarrow A=\dfrac{1}{2}-\dfrac{1}{2.3^{99}}< \dfrac{1}{2}\)
HH
2
2 tháng 8 2023
a: 2A=2^2+2^3+...+2^21
=>A=2^21-2
b: B=2+2^2+...+2^100
=>2B=2^2+2^3+...+2^101
=>B=2^101-2
c: C=3+3^2+...+3^10
=>3C=3^2+3^3+...+3^11
=>2C=3^11-3
=>C=(3^11-3)/2
KS
2 tháng 8 2023
`A = 2 + 2^2 + ... + 2^20`
`=> 2A = 2^2 + 2^3 + ... +2^21`
`=> 2A-A = (2^2 + 2^3 + ... + 2^21) - (2 + 2^2 + ... +2^20)`
`=> A = 2^21 - 2`
`B = 2 + 2^2 + ... + 2^99 + 2^100`
`=>2B = 2^2 + 2^3 + ... + 2^100 + 2^101`
`=> 2B-B = (2^2 + 2^3 + ... + 2^101)- (2 + 2^2 + ... + 2^100)`
`=> B = 2^101 - 2`
`C = 3 + 3^2 + .... + 3^10`
`=>3C = 3^2 + 3^3 + ... +3^11`
`=>3C - C = (3^2 + 3^3 + ... +3^11) - (3 + 3^2 + .... + 3^10)`
`=> 2C = 3^11 - 3`
`=> C = (3^11 - 3)/2
12 tháng 10 2023
a: \(12+2^2+3^2+4^2+5^2\)
\(=12+4+9+16+25\)
\(=16+50=66\)
\(\left(1+2+3+4+5\right)^2=15^2=225\)
=>\(12+2^2+3^2+4^2+5^2< \left(1+2+3+4+5\right)^2\)
b: \(1^3+2^3+3^3+4^3=\left(1+2+3+4\right)^2< \left(1+2+3+4\right)^3\)
c: \(5^{202}=5^2\cdot5^{200}=25\cdot5^{200}>16\cdot5^{200}\)
d: \(18\cdot4^{500}=18\cdot2^{1000}\)
\(2^{1004}=2^4\cdot2^{1000}=16\cdot2^{1000}\)
=>\(18\cdot4^{500}>2^{1004}\)
e: \(2022\cdot2023^{2024}+2023^{2024}=2023^{2024}\left(2022+1\right)\)
\(=2023^{2025}\)
21 tháng 12 2020
2.
a, x-13=-46
=>x=(-46)+13
=>x=33
b, 4x-6=22
=>4x=22+6
=>4x=28
=>x=28:4
=>x=7
3.
a, 32=25
48=24.3
=>ƯCLN(32,48)=24=16
16=24
72=23.32
=>ƯCLN(16,72)=23=8
b,
24=23.3
60=22.3.5
=>BCNN(24,60)=23.3.5=120
72=23.32
180=22.32.5
=>BCNN(72,180)=23.32.5=360
13 tháng 11 2023
1: \(A=2+2^2+2^3+2^4+...+2^{97}+2^{98}+2^{99}+2^{100}\)
\(=2\left(1+2+2^2+2^3\right)+...+2^{97}\left(1+2+2^2+2^3\right)\)
\(=15\left(2+2^5+...+2^{97}\right)\)
\(=30\left(1+2^4+...+2^{96}\right)⋮30\)
2:
\(B=3+3^2+3^3+...+3^{2022}\)
\(=\left(3+3^2\right)+\left(3^3+3^4\right)+...+\left(3^{2021}+3^{2022}\right)\)
\(=\left(3+3^2\right)+3^2\left(3+3^2\right)+...+3^{2020}\left(3+3^2\right)\)
\(=12\left(1+3^2+...+3^{2020}\right)⋮12\)
DH
4
\(2^{32}< 2^{33}=\left(2^3\right)^{11}=8^{11}\)
\(3^{22}=\left(3^2\right)^{11}=9^{11}\)
\(\Rightarrow9^{11}>8^{11}\Rightarrow3^{22}>2^{32}\)