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a) 3^200 và 2^300
ta có:3^200=3^2x100=(3^2)^100=9^100
2^300=2^3x100=(2^3)^100=8^100
vì 9>8 =>9^100>8^100
=>3^200>2^200
vậy...
b)125^5 và 25^7
ta có:125^5=(5^3)^5=5^15
25^7=(5^2)^7=5^14
vì 15>14 =>5^15>5^14
=>125^5>25^7
vậy.....
c)9^20 và 27^13
ta có:9^20=(3^2)^20=3^40
27^13=(3^3)^13=3^39
vì 40>39 => 3^40>3^39
=>9^20>27^13
vậy....
d)3^54 và 2^81
ta có:3^54=3^6x9=(3^6)^9=729^9
2^81=2^9x9=(2^9)^9=512^9
vì 729>512 =>729^9>512^9
=> 3^54>2^81
vậy.....
e)10^30 và 2^100
ta có: 10^30=10^3x10=(10^3)^10=1000^10
2^100=2^10x10=(2^10)^10=1024^10
vì 1000<1024 =>1000^10<1024^10
=> 10^30<2^100
vậy....
f)5^40 và 620^10
ta có:5^40=5^4x10=(5^4)^10=625^10
vì 625>620 =>625^10>620^10
=>5^40>620^10
vậy....
ĐÓ LÀ CÁCH LÀM CỦA TỚ NẾU THẤY ĐÚNG THÌ K NHA.
a) 3^200 = (3^2)^100 = 9^100
2^300 = (2^3)^100 = 8 ^100
Do 9>8 =>9^100 > 8^100=> 3^200> 2^300
b) 125^5 = (5^3)5 = 5^15
25^7 = ( 5^2)^7 = 5^14
Do 5^15 > 5^14 => 125^5 > 25^7
Trước tiên ta nên tìm kết quả :
=> có 99 số số hạng
Tổng của kết quả đó là :
( 99 + 1 ) . 99 : 2 = 4950
Vậy ta có : 2-(x+3) = 4950
x+3 = 2 - 4950
x+3 = -4948
x = -4948 - 3
x = -4951
2-x-3 = (1+99) +(2+98)+...+( 49+51)+ 50
-1-x = 10+10 +..+ 10 + 50
-1-x = 490+50
-x= 540 + 1
-x = 541
=> x= -541
Từ 2 giả thiết: \(a+b+c=2018;\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}=\frac{6}{2018}\)
\(\Rightarrow\left(a+b+c\right).\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)=\frac{2018.6}{2018}=6\)
\(\Leftrightarrow\frac{a+b+c}{a+b}+\frac{a+b+c}{b+c}+\frac{a+b+c}{c+a}=6\)
\(\Leftrightarrow1+\frac{c}{a+b}+1+\frac{a}{b+c}+1+\frac{b}{c+a}=6\)
\(\Leftrightarrow\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}=3\)
Vậy giá trị của biểu thức đó là 3.
\(\frac{2}{2.3}\) + \(\frac{2}{3.4}\) + \(\frac{2}{4.5}\) + .......+ \(\frac{2}{x.\left(x+1\right)}\) = \(\frac{2017}{2019}\)
2 . ( \(\frac{1}{2}\) - \(\frac{1}{3}\) + \(\frac{1}{3}\) - \(\frac{1}{4}\) + .......+ \(\frac{1}{x+1}\) ) = \(\frac{2017}{2019}\)
2 . ( \(\frac{1}{2}\) - \(\frac{1}{x+1}\) ) = \(\frac{2017}{2019}\)
\(\frac{1}{2}\) - \(\frac{1}{x+1}\) = \(\frac{2017}{2019}\) : 2
\(\frac{1}{2}\) - \(\frac{1}{x+1}\) = \(\frac{2017}{4038}\)
\(\frac{1}{x+1}\) = \(\frac{1}{2}\) - \(\frac{2017}{4038}\)
\(\frac{1}{x+1}\) = \(\frac{1}{2019}\)
<=> x + 1 = 2019 => x = 2018
vậy x = 2018
\(\frac{2}{2.3}+\frac{2}{3.4}+\frac{2}{4.5}+...+\frac{2}{x\left(x+1\right)}=\frac{2017}{2019}\)
\(\Leftrightarrow2\left(\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{x\left(x+1\right)}\right)=\frac{2017}{2019}\)
\(\Leftrightarrow2\left(\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{x}-\frac{1}{x+1}\right)=\frac{2017}{2019}\)
\(\Leftrightarrow2\left(\frac{1}{2}-\frac{1}{x+1}\right)=\frac{2017}{2019}\)
\(\Leftrightarrow\frac{1}{2}-\frac{1}{x+1}=\frac{2017}{4038}\)
\(\Leftrightarrow\frac{1}{x+1}=\frac{1}{2019}\)
\(\Rightarrow x+1=2019\)
\(\Leftrightarrow x=2018\)
Vậy \(x=2018\)
1) ta có:\(2^{150}\)= (2^3)^50=8^50
\(3^{100}\)= (3^2)^50 = 9^50
vì 8^50 < 9^50 => \(2^{150}\)<\(3^{100}\)
\(\dfrac{1}{38}>\dfrac{1}{40}>\dfrac{1}{42}>...>\dfrac{1}{50}\)
=>\(\dfrac{1}{38}+\dfrac{1}{40}+\dfrac{1}{42}+\dfrac{1}{44}+\dfrac{1}{46}+\dfrac{1}{48}+\dfrac{1}{50}< 7\cdot\dfrac{1}{38}=\dfrac{7}{38}< 1\)
Vậy tổng trên bé hơn 1
A=-1-3-5-...-2017
=-(1+3+5+...+2017)
Xét tổng B=1+3+5+...+2017
Tổng B có:(2017-1):2+1=1009(số hạng)
Tổng B=\(\dfrac{\left(2017+1\right)\cdot1009}{2}=1009\cdot1009=1018081\)
=>A=-B=-1018081
Ta có
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2018^2}\) < \(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2017.2018}\)
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2018^2}\)< 1 - \(\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2017}-\frac{1}{2018}\)
\(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2018^2}\)< 1 - \(\frac{1}{2018}\)= \(\frac{2017}{2018}\)< 1
Vậy \(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2018^2}\)< 1 ( dpcm )
Ta có:
\(\frac{1}{2^2}\)< \(\frac{1}{1.2}\).
\(\frac{1}{3^2}\)< \(\frac{1}{2.3}\).
\(\frac{1}{4^2}\)< \(\frac{1}{3.4}\).
...
\(\frac{1}{2017^2}\)< \(\frac{1}{2016.2017}\).
\(\frac{1}{2018^2}\)< \(\frac{1}{2017.2018}\).
Từ trên ta có:
\(\frac{1}{2^2}\)+ \(\frac{1}{3^2}\)+ \(\frac{1}{4^2}\)+...+ \(\frac{1}{2017^2}\)+ \(\frac{1}{2018^2}\)< \(\frac{1}{1.2}\)+ \(\frac{1}{2.3}\)+ \(\frac{1}{3.4}\)+...+ \(\frac{1}{2016.2017}\)+ \(\frac{1}{2017.2018}\)= 1- \(\frac{1}{2}\)+ \(\frac{1}{2}\)- \(\frac{1}{3}\)+ \(\frac{1}{3}\)- \(\frac{1}{4}\)+...+ \(\frac{1}{2016}\)- \(\frac{1}{2017}\)+ \(\frac{1}{2017}\)- \(\frac{1}{2018}\)= 1- \(\frac{1}{2018}\)< 1.
=> \(\frac{1}{2^2}\)+ \(\frac{1}{3^2}\)+ \(\frac{1}{4^2}\)+...+ \(\frac{1}{2017^2}\)+ \(\frac{1}{2018^2}\)< 1.
=> ĐPCM.