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NK
1
6 tháng 3 2023
\(A=\left(\dfrac{456}{2}+1\right)+...+\left(\dfrac{2}{456}+1\right)+\left(\dfrac{1}{457}+1\right)+1\)
\(A=458+\dfrac{458}{2}+....+\dfrac{458}{456}+\dfrac{458}{457}-\dfrac{458}{458}\)
\(A=458\left(\dfrac{1}{2}+...+\dfrac{1}{456}+\dfrac{1}{457}+\dfrac{1}{458}\right)\)
Ta xét \(\dfrac{1}{2}+....+\dfrac{1}{456}+\dfrac{1}{457}+\dfrac{1}{458}\)có :
\(\dfrac{1}{2}=\dfrac{1}{2}\)
\(\dfrac{1}{3}+\dfrac{1}{4}>\dfrac{1}{4}+\dfrac{1}{4}=\dfrac{1}{2}\)
\(\dfrac{1}{5}+\dfrac{1}{6}+...+\dfrac{1}{8}>\dfrac{1}{8}+\dfrac{1}{8}+...+\dfrac{1}{8}=\dfrac{1}{2}\)
\(\dfrac{1}{9}+\dfrac{1}{10}+....+\dfrac{1}{16}>\dfrac{1}{16}+....+\dfrac{1}{16}=\dfrac{1}{2}\)
\(\dfrac{1}{17}+\dfrac{1}{18}+....+\dfrac{1}{32}>\dfrac{1}{32}+.....+\dfrac{1}{32}=\dfrac{1}{2}\)
\(\dfrac{1}{33}+\dfrac{1}{34}+....+\dfrac{1}{64}>\dfrac{1}{64}+....+\dfrac{1}{64}=\dfrac{1}{2}\)
\(\dfrac{1}{65}+\dfrac{1}{66}+.....+\dfrac{1}{128}>\dfrac{1}{128}+....+\dfrac{1}{128}=\dfrac{1}{2}\)
\(\dfrac{1}{129}+\dfrac{1}{130}+.....+\dfrac{1}{256}>\dfrac{1}{256}+....+\dfrac{1}{256}=\dfrac{1}{2}\)
\(\dfrac{1}{257}+\dfrac{1}{258}+....+\dfrac{1}{458}>\dfrac{1}{458}+...+\dfrac{1}{458}=\dfrac{1}{2}\)
Vậy ta thấy được rằng
\(\dfrac{1}{2}+...+\dfrac{1}{456}>\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{1}{2}+\dfrac{202}{458}\)
\(=4+\dfrac{202}{458}=\dfrac{2034}{458}\)
Vậy \(A>458.\dfrac{2034}{458}=2034\)
Hay tức là A > 2016 ( đpcm )
DP
2
MP
23 tháng 3 2022
Ta có:
A = (457/1 + 1) + (456/2 + 1) + ... + (2/456 + 1) + (1/457 + 1) - 457
A = 458 + 458/2 + ... + 458/456 + 458/457 - 457
A = 458 (1 + 1/2 + ...+ 1/456 + 1/457) - 457
Xét 1 + 1/2 + ... + 1/456 + 1/457, ta có
1 = 1
1/2 = 1/2
1/3 + 1/4 > 1/4 + 1/4 = 1/2
1/5 + 1/6 + ... + 1/8 > 1/8 + 1/8 + ... + 1/8 = 1/2
1/9 + 1/10 +...+ 1/16 > 1/16 + 1/16 +...+ 1/16 = 1/2
1/17 + 1/18 + ... + 1/32 > 1/32 + ... + 1/32 = 1/2
1/33+ 1/34 + ... + 1/64 > 1/64 + ...+ 1/64 = 1/2
1/65 + 1/66 + ...+ 1/128 > 1/128 + ... + 1/128 = 1/2
1/129 + 1/130 + ... + 1/256 > 1/256 + ...+ 1/256 = 1/2
1/257 + 1/258 + ... + 1/457 > 1/457 + ... + 1/457 = 201/457 > 0,4
Vậy 1 + 1/2 + ... + 1/456 + 1/457 > 1 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 0,4 = 5,4
Vậy A > 458*5,4 - 457 = 2016,2
Vậy A > 2016.
MP
23 tháng 3 2022
Ta có:
A = (456/2 + 1) + ... + (2/456 + 1) + (1/457 + 1) + 1
A = 458 + 458/2 + ... + 458/456 + 458/457 - 458/458
A = 458 (1/2 + ...+ 1/456 + 1/457 + 1/458)
Xét 1/2 + ... + 1/456 + 1/458, ta có
1/2 = 1/2
1/3 + 1/4 > 1/4 + 1/4 = 1/2
1/5 + 1/6 + ... + 1/8 > 1/8 + 1/8 + ... + 1/8 = 1/2
1/9 + 1/10 +...+ 1/16 > 1/16 + 1/16 +...+ 1/16 = 1/2
1/17 + 1/18 + ... + 1/32 > 1/32 + ... + 1/32 = 1/2
1/33+ 1/34 + ... + 1/64 > 1/64 + ...+ 1/64 = 1/2
1/65 + 1/66 + ...+ 1/128 > 1/128 + ... + 1/128 = 1/2
1/129 + 1/130 + ... + 1/256 > 1/256 + ...+ 1/256 = 1/2
1/257 + 1/258 + ... + 1/458 > 1/458 + ... + 1/458 = 202/458
Vậy 1/2 + ... + 1/456 + 1/457 > 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 1/2 + 202/458 = 4 + 202/458 = 2034/458
Vậy A > 458*2034/458 = 2034
Vậy A > 2016.
TN
8 tháng 4 2016
đặt B=1/2.3+1/3.4+...+1/49.50
=1/1.2+1/2.3+1/3.4+...+1/49.50
=1-1/2+1/2-1/3+...+1/49-1/50
=1-1/50<1 (1)
Mà 1<2(2)
A =1/1+1/2.2+1/3.3+...+1/50.50<1-1/2+1/2-1/3+...+1/49-1/50 (3)
từ (1),(2),(3) =>A<2
BM
8 tháng 4 2016
Ta có : \(A=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+...........+\frac{1}{50^2}=1+\frac{1}{2^2}+........+\frac{1}{50^2}\)
=> \(A<1+\frac{1}{1.2}+\frac{1}{2.3}+.............+\frac{1}{49.50}\)
=> \(A<1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+.........+\frac{1}{49}-\frac{1}{50}\)
=> \(A<2-\frac{1}{50}\Rightarrow A<2\)
Vậy A nhỏ hơn 2
TP
1
TP
25 tháng 4 2017
\(A=\frac{2016^{2016}+2}{2016^{2016}-1};;B=\frac{2016^{2016}}{2016^{2016}-3}\)\(A=\frac{\left(2016^{2016}-1\right)+2+1}{2016^{2016}-1};;B=\frac{\left(2016^{2016}-3\right)+3}{2016^{2016}-3}\)\(A=1+\frac{3}{2016^{2016}-1};;B=1+\frac{3}{2016^{2016}-3}\);;Vì \(2016^{2016}-1>2016^{2016}-3\)Nên\(\frac{3}{2016^{2016}-1}< \frac{3}{2016^{2016}-3}\)Vậy \(A< B\)
NN
0
9 tháng 8 2016
\(A=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{2015^2}+\frac{1}{2016^2}\)
\(A< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{2014.2015}+\frac{1}{2015.2016}\)
\(A< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{2014}-\frac{1}{2015}+\frac{1}{2015}-\frac{1}{2016}\)
\(A< 1-\frac{1}{2016}\)
\(A< \frac{2015}{2016}\left(đpcm\right)\)
9 tháng 8 2016
\(A=\frac{1}{2.2}+\frac{1}{3.3}+.....+\frac{1}{2016.2016}< \frac{1}{1.2}+\frac{1}{2.3}+.....+\frac{1}{2015.2016}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-.....+\frac{1}{2015}-\frac{1}{2016}\)
\(=1-\frac{1}{2016}\)
\(=\frac{2015}{2016}\)
\(\Rightarrow A< \frac{2015}{2016}\)
ta có: A=457/1+456/2+455/3+...+1/457
suy ra: A= 457/1+1+456/2+1+...+1/457+1-467
suy ra: A=458/1+458/2+458/3+...+458/457 -457
suy ra A=458(1+1/2+1/3+...+1/457)-457
suy ra :A=458 * 6,702992749 -457
suy ra: A>2291>2016
suy ra: A>2016
vậy A>2016