Cho A=3^101 + 3^102 +3^103 + ..... + 3^200
CMR : A chia hết cho 120
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LM
0
I
3
2 tháng 8 2021
a) \(3^{10}+3^{11}+3^{12}\)
⇔ \(3^{10}\left(1+3+3^2\right)\)
⇔ \(3^{10}.13\)
⇒ \(3^{10}.13\) chia hết cho 13
24 tháng 4 2016
\(B=\frac{101}{102}+\frac{102}{103}+\frac{103}{101}\)
\(B=1\)
B < 3
22 tháng 2 2020
a) (-1 + 2 - 3 + 4 -...- 49 + 50 ) - ( 1 - 2 + 3 - 4 +...+ 49 - 50)
= -1 + 2 - 3 + 4 -...- 49 + 50 - 1 + 2 -3 + 4 -... - 49 + 50)
=-1 -1
=-1 + (-1)
=-2
Mình nghĩ là đúng đó ,mình nên nhìn kĩ B1 và B2
b) Tự làm nhé
22 tháng 2 2020
2)
a) (a - 3) - (a - 5)
=a - 3 - a + 5
=a - a - 3 + 5
= 0 - (3 - 5)
= -(3 - 5)
= - (-2) =2
b) ( a + b - c) - (a - c)
=a + b - c - a + c
= a - a + b - c +c
= 0 + b + c - c
= b + ( c - c)
= b + 0
= b
c) ( a + b ) - ( a - c -d + b)
= a + b - a + c +d -b )
= a - a + (b -b) - c + d
= 0 + 0 - c+d
= 0 - c + d
= - c+d
TT
0
TA
1
2 tháng 8 2023
\(A=\dfrac{1}{101}+\dfrac{1}{102}+\dfrac{1}{103}+...+\dfrac{1}{199}+\dfrac{1}{120}\left(a\right)\)
\(\Rightarrow A=\left(\dfrac{1}{101}+\dfrac{1}{102}+...\dfrac{1}{125}\right)+\left(\dfrac{1}{126}+\dfrac{1}{127}+...\dfrac{1}{150}\right)+\left(\dfrac{1}{151}+\dfrac{1}{152}+...\dfrac{1}{175}\right)+\left(\dfrac{1}{176}+\dfrac{1}{177}+...\dfrac{1}{200}\right)\)
\(\Rightarrow A>25.\dfrac{1}{125}+25.\dfrac{1}{150}+25.\dfrac{1}{175}+25.\dfrac{1}{200}\)
\(\Rightarrow A>\dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}\)
\(\Rightarrow A>\dfrac{168+140+120+105}{840}=\dfrac{533}{840}>\dfrac{5}{8}\left(\dfrac{533}{840}>\dfrac{525}{840}\right)\)
\(\Rightarrow A>\dfrac{5}{8}\left(1\right)\)
\(\left(a\right)\Rightarrow A=\left(\dfrac{1}{101}+...\dfrac{1}{120}\right)+\left(\dfrac{1}{121}+...\dfrac{1}{140}\right)+\left(\dfrac{1}{141}+...\dfrac{1}{160}\right)+\left(\dfrac{1}{161}+...\dfrac{1}{180}\right)+\left(\dfrac{1}{181}+...\dfrac{1}{200}\right)\)
\(\Rightarrow A< 20.\dfrac{1}{100}+20.\dfrac{1}{120}+20.\dfrac{1}{140}+20.\dfrac{1}{160}+20.\dfrac{1}{180}\)
\(\Rightarrow A< \dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{1}{9}\)
\(\Rightarrow A< \dfrac{1}{5}+\dfrac{1}{6}+\dfrac{1}{7}+\dfrac{1}{8}+\dfrac{504+420+360+315+280}{2520}=\dfrac{1879}{2520}< \dfrac{3}{4}\left(\dfrac{1879}{2520}< \dfrac{1890}{2520}\right)\)
\(\Rightarrow A< \dfrac{3}{4}\left(2\right)\)
\(\left(1\right),\left(2\right)\Rightarrow\dfrac{5}{8}< A< \dfrac{3}{4}\left(dpcm\right)\)
DK
1
20 tháng 6 2015
(1-2+3-4+5-6+7-8+...+101-102+103) * x-120=2012
[ -1 + ( -1 ) ... ( -1 ) + 103] * x -120 = 2012
( -1 x 51 + 103 ) * x - 120 = 2012
( -51 +103) * x = 2012 + 120
52 * x =2132
x= 2132 : 52
x= 41
\(A=3^{101}+3^{102}+3^{103}+...+3^{200}\)
\(3A=3^{102}+3^{103}+3^{104}+...+3^{201}\)
\(3A-A=\left(3^{102}+3^{103}+3^{104}+3^{201}\right)-\left(3^{101}+3^{102}+3^{103}+...+3^{201}\right)\)
\(2A=3^{201}-3^{101}\)
\(2A=3^{100}\)
\(\Rightarrow A=3^{100}:2\)
\(A=3^{101}+3^{102}+3^{103}+...+3^{200}\)
\(A=3^{101}+3^{102}+3^{103}+3^{104}+...+3^{197}+3^{198}+3^{199}+3^{200}\)
\(A=3^{100}\left(3+3^2+3^3+3^4\right)+...+3^{196}\left(3+3^2+3^3+3^4\right)\)
\(A=120\left(3^{100}+...+3^{196}\right)⋮120\)