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Câu hỏi của Thùyy Linhh - Toán lớp 6 - Học toán với OnlineMath

Ta có:

\(C=\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot\cdot\cdot\cdot\cdot\frac{9999}{10000}\)

Đặt \(I=\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}\cdot\cdot\cdot\cdot\cdot\frac{10000}{10001}\)

Ta có: \(\frac{1}{2}< \frac{2}{3};\frac{3}{4}< \frac{4}{5};\frac{5}{6}< \frac{6}{7};.....;\frac{9999}{10000}< \frac{10000}{10001}\)

\(\Rightarrow C< D\)

Lại có: \(C\cdot D=\left(\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot\cdot\cdot\cdot\cdot\frac{9999}{10000}\right)\cdot\left(\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}\cdot\cdot\cdot\cdot\cdot\frac{10000}{10001}\right)\)

\(\Leftrightarrow C\cdot D=\frac{1}{2}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot\cdot\cdot\cdot\cdot\frac{9999}{10000}\cdot\frac{2}{3}\cdot\frac{4}{5}\cdot\frac{6}{7}\cdot\cdot\cdot\cdot\cdot\frac{10000}{10001}\)

\(\Leftrightarrow C\cdot D=\frac{1}{2}\cdot\frac{2}{3}\cdot\frac{3}{4}\cdot\frac{5}{6}\cdot\frac{6}{7}\cdot\cdot\cdot\cdot\cdot\frac{9999}{10000}\cdot\frac{10000}{10001}\)

\(\Leftrightarrow C\cdot D=\frac{1}{10001}\)

Mà C<D \(\Rightarrow C\cdot C< C\cdot D\)

Hay \(C\cdot C< \frac{1}{10001}\)

\(\Rightarrow C< \frac{1}{10001}< \frac{1}{100}\)

Vậy \(C< \frac{1}{100}\left(đpcm\right)\)

Đặt :\(C=\frac{1}{2}.\frac{3}{4}.\frac{5}{6}.....\frac{9999}{10000}\)

\(N=\frac{2}{3}.\frac{4}{5}...\frac{10000}{10001}\)

Ta thấy:\(\frac{1}{2}< \frac{2}{3};\frac{3}{4}< \frac{4}{5};....;\frac{9999}{10000}< \frac{10000}{10001}\)

Mặt khác ta thấy:

\(C.N=\left(\frac{1}{2}.\frac{3}{4}.\frac{5}{6}...\frac{9999}{10000}\right).\left(\frac{2}{3}.\frac{4}{5}.\frac{6}{7}...\frac{10000}{10001}\right)\)

\(C.N=\frac{1}{2}.\frac{2}{3}.\frac{3}{4}....\frac{9999}{10000}.\frac{10000}{10001}\)

\(C.N=\frac{1.2.3....9999.10000}{2.3.4....10000.10001}\)

Rút gọn  phép tính \(C.N\)

\(C.N=\frac{1}{10001}\)

\(C.C< N\Rightarrow C.C< C.N\)

Hay\(C.C< \frac{1}{10001}< \frac{1}{10000}=\frac{1}{10}.\frac{1}{10}\)

\(\Rightarrow C< \frac{1}{10000}\)(đpcm)

B< 1+(1/1.2+1/2.3+...+1/62.63)

B<1+(1-1/2+1/2-1/3+...+1/62-1/63)

B<1+1-1/63

B<2-1/63

B<6-3/189

mà 6-3/189<6 

Vậy B<6

b, gọi D=2/3.4/5....10000/10001

Ta có: 1/2<2/3     3/4<4/5      .. .....      9999/10000<10000/10001

=> C<D                 1

C.D=1/2.3.4.....9999/10000.2/3.4/5...10000/10001

C.D=1/10001       2

Từ 1 : C<D => C.C<C.D<1/10001

                   =>C^2<1/10001<1/10000

                   =>C^2<(1/100)^2

Vậy C<1/100 (đpcm)

a, Ta có : \(\dfrac{1}{2^2}=\dfrac{1}{4};\dfrac{1}{3^2}< \dfrac{1}{2.3}=\dfrac{1}{2}-\dfrac{1}{3};\dfrac{1}{4^2}< \dfrac{1}{3.4}=\dfrac{1}{3}-\dfrac{1}{4}\)

\(...\dfrac{1}{100^2}< \dfrac{1}{99.100}=\dfrac{1}{99}-\dfrac{1}{100}\)

\(A=\dfrac{1}{4}+\dfrac{1}{2}-\dfrac{1}{100}< 2\)

@Nguyễn Khanh

b, 1 = 1
1/2 + 1/3 = 1/(1 + 1) + 1/(1 + 2) < 2/(1 + 1) = 2/2 = 1
1/4 + 1/5 + 1/6 + 1/7 = 1/(3 + 1) + 1/(3 + 2) + 1/(3 + 3) + 1/(3 + 4) < 4/(3 + 1) = 4/4 = 1
1/8 + 1/9 + ... + 1/15 = 1/(7 + 1) + 1/(7 + 2) + ... + 1/(7 + 8) < 8/(7 + 1) = 8/8 = 1
1/16 + 1/17 + ... + 1/31 = 1/(15 + 1) + 1/(15 + 2) + ... + 1/(15 + 16) < 16/(15 + 1) = 16/16 = 1
1/32 + 1/33 + ... + 1/63 = 1/(31 + 1) + 1/(31 + 2) + ... + 1/(31 + 32) < 32/(31 + 1) = 32/32 = 1
=> 1 + 1/2 + 1/3 + 1/4 + ... + 1/64 < 1 + 1 + 1 + 1 + 1 + 1
=> 1 + 1/2 + 1/3 + 1/4 + ... + 1/64 < 6 (đpcm)
@Nguyễn Khanh