CMR : 1/4 + 1/16 + 1/36 + ... + 1/576 < 1/2
Chứng tỏ rằng:A=1/4+1/16+1/36+1/64+...+1/576<1/2
THAM KHẢO NHA
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Chứng minh rằng: A=1/4+1/16+1/36+1/64+...+1/576<1/2
Chứng minh A = 1/4 + 1/6 + 1/36 + ....... + 1/576 < 1/2
CMR:
1/4+1/16+1/36+....+1/10000<1/2
Đặt \(M=\frac{1}{4}+\frac{1}{16}+.....+\frac{1}{10000}\)
\(M=\frac{1}{2.2}+\frac{1}{4.4}+\frac{1}{6.6}+......+\frac{1}{100.100}\)
\(=\frac{1}{2.2}\left(1+\frac{1}{2.2}+\frac{1}{3.3}+.....+\frac{1}{50.50}\right)\)
\(< \frac{1}{2.2}\left(1+\frac{1}{1.2}+\frac{1}{2.3}+.....+\frac{1}{49.50}\right)\)
\(=\frac{1}{4}.\left(1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+....+\frac{1}{49}-\frac{1}{50}\right)\)
\(=\frac{1}{4}.\left(1+1-\frac{1}{50}\right)< \frac{1}{4}.\left(1+1\right)=\frac{1}{4}.2=\frac{1}{2}\)
Vậy \(M< \frac{1}{2}\)
CMR : 1/4 + 1/16 + 1/36 + 1/64 + 1/100 + 1/144 + ... + 1/10000 < 1/2
\(Đ\text{ặt }S=\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+....+\frac{1}{10000}\)
\(S=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+...+\frac{1}{100^2}\)
\(S=\frac{1}{2^2}\cdot\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)\)
Ta có :
\(\frac{1}{2^2}< \frac{1}{1\cdot2};\text{ }\frac{1}{3^2}< \frac{1}{2\cdot3};\text{ }...;\text{ }\frac{1}{50^2}< \frac{1}{49\cdot50}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< \frac{1}{1\cdot2}+\frac{1}{2\cdot3}+...+\frac{1}{49\cdot50}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)
\(=1-\frac{1}{50}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< 1\Rightarrow1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}< 1+1=2\)
\(\Rightarrow\frac{1}{2^2}\left(1+\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{50^2}\right)< \frac{1}{2^2}\cdot2\)
\(\Rightarrow S< \frac{1}{2}\) (ĐPCM)
Đặt \(A=\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+....+\frac{1}{10000}\)
\(\Rightarrow A=\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{100^2}\)
\(\Rightarrow4A=1+\frac{1}{2^2}+\frac{1}{3^2}+....+\frac{1}{50^2}\)
\(\Rightarrow4A< 1+\frac{1}{1.2}+\frac{1}{2.3}+....+\frac{1}{49.50}\)
\(\Rightarrow4A=1+1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{49}-\frac{1}{50}\)
\(\Rightarrow4A< 2-\frac{1}{50}< 2\)
\(\Rightarrow4A< 2\Rightarrow A< \frac{2}{4}=\frac{1}{2}\)
=>a<1/2
chung to A=1/4+1/16+.....+1/576<1/2
\(A=\frac{1}{4}+\frac{1}{16}+....+\frac{1}{576}\)
\(\Rightarrow4A=1+\frac{1}{4}+....+\frac{1}{144}\)
\(\Rightarrow4A-A=1-\frac{1}{576}\)
\(\Rightarrow3A=1-\frac{1}{576}\)
\(\Rightarrow A=\frac{1-\frac{1}{576}}{3}\)
Ta có: \(\frac{1-\frac{1}{576}}{3}< \frac{1}{3}< \frac{1}{2}\)
vậy suy ra đpcm
A=14 +116 +....+1576
⇒4A=1+14 +....+1144
⇒4A−A=1−1576
⇒3A=1−1576
⇒A=1−1576 3
Ta có: 1−1576 3 <13 <12
vậy suy ra đpcm
S = 1/4 + 1/9 + 1/16 + 1/25 + 1/36 + 1/49 + 1/64 + 1/81
CMR: 2/2 < S < 8/9
S=1/4+1/9+1/16+1/25+1/36+1/49+1/64+1/81=1-1/81=1/81
vô lí vì 2/2 = 1 mà 8/9 < 1
(1 - 1/4)(1 - 1/9)(1 - 1/16) *...(1- 1 576 )*(1- 1 625 )
Tính:
A=1*2*3+2*3*4+3*4*5+......+98*99*100
CMR:
A=1/4+1/16+1/36+1/64+1/100+1/144+1/196<1/2
A=\(\frac{1}{4}+\frac{1}{16}+\frac{1}{36}+\frac{1}{64}+\frac{1}{100}+\frac{1}{144}+\frac{1}{196}\)=\(\frac{1}{2^2}+\frac{1}{4^2}+\frac{1}{6^2}+\frac{1}{8^2}+\frac{1}{10^2}+\frac{1}{12^2}+\frac{1}{14^2}\)
=>A<\(\frac{1}{2.2}+\frac{1}{2.4}+\frac{1}{4.6}+\frac{1}{6.8}+\frac{1}{8.10}+\frac{1}{10.12}+\frac{1}{12.14}\)
=>A<\(\left(\frac{1}{2}-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{6}+...+\frac{1}{12}-\frac{1}{14}\right)\)\(:2\)=\(\left(\frac{1}{2}-\frac{1}{14}\right):2\)<\(\frac{1}{2}\)
=>A<\(\frac{1}{2}\)