cho A=\(\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{899}{900}\)
chứng minh A không phải là số nguyên
ai giúp mk tick cho
K
Khách
Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
Những câu hỏi liên quan
PT
1
7 tháng 5 2017
\(A=\frac{3}{4}+\frac{8}{9}+\frac{15}{16}+...+\frac{899}{900}\)
\(A=\left(1-\frac{1}{4}\right)+\left(1-\frac{1}{9}\right)+\left(1-\frac{1}{16}\right)+...+\left(1-\frac{1}{900}\right)\)
\(A=\left(1+1+1+...+1\right)-\left(\frac{1}{4}+\frac{1}{9}+\frac{1}{16}+...+\frac{1}{900}\right)\)
\(A=29-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{30^2}\right)\)
đặt \(B=\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{30^2}\)
Ta thấy \(\frac{1}{2^2}< \frac{1}{1.2}\); \(\frac{1}{3^2}< \frac{1}{2.3}\); \(\frac{1}{4^2}< \frac{1}{3.4}\); ... ; \(\frac{1}{30^2}< \frac{1}{29.30}\)
\(\Rightarrow B< \frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{29.30}\)
\(=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{29}-\frac{1}{30}\)
\(=1-\frac{1}{30}< 1\)
\(\Rightarrow B< 1\)
\(\Rightarrow A=29-\left(\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+...+\frac{1}{30^2}\right)< 29\)
6 tháng 5 2016
A=\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.........\frac{899}{900}\)
A=\(\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}..........\frac{29.31}{30.30}\)
A=\(\frac{1.2.3.......29}{2.3.4.......30}.\frac{3.4.5........31}{2.3.4.......30}\)
A=\(\frac{1}{30}.\frac{2}{31}=\frac{1}{465}\)
18 tháng 3 2018
\(A=\frac{3}{4}.\frac{8}{9}.........\frac{899}{900}\)
\(=\frac{1.3}{2^2}.\frac{2.4}{3^2}.....\frac{29.31}{30^2}=\frac{1.2....29}{2.3....30}.\frac{3.4....31}{2.3....30}\)
\(=\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)
17 tháng 8 2016
B = 3/4 + 8/9 + 15/16 + .... + 2499/2500
B = (1 - 1/4) + (1 - 1/9) + (1 - 1/16) + ... + (1 - 1/2500)
B = (1 - 1/22) + (1 - 1/32) + (1 - 1/42) + ... + (1 - 1/502)
B = (1 + 1 + 1 + ... + 1) - (1/22 + 1/32 + 1/42 + ...+ 1/502)
49 số 1
B = 49 - (1/22 + 1/32 + 1/42 + ... + 1/502)
=> B < 49 (1)
B > 49 - (1/1×2 + 1/2×3 + 1/3×4 + ... + 1/49×50)
B > 49 - (1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/49 - 1/50)
B > 49 - (1 - 1/50)
B > 49 - 1 + 1/50
B > 48 + 1/50 > 48 (2)
Từ (1) và (2) => 48 < B < 49
=> B không phải là số nguyên ( đpcm)
17 tháng 8 2016
B = 3/4 + 8/9+ 15/16 + ... + 2499/2500
B = (1 - 1/4) + (1 - 1/9) + (1 - 1/16) + ... + (1 - 1/2500)
B = (1 - 1/22) + (1 - 1/32) + (1 - 1/42) + ... + (1 - 1/502)
B = (1 + 1 + 1 + ... + 1) - (1/22 + 1/32 + 1/42 + .... + 1/502)
49 số 1
=> B = 49 - (1/22 + 1/32 + 1/42 + ... + 1/502)
=> B < 49 (1)
B > 49 - (1/1×2 + 1/2×3 + 1/3×4 + ... + 1/49×50)
B > 49 - (1 - 1/2 + 1/2 - 1/3 + 1/3 - 1/4 + ... + 1/49 - 1/50)
B > 49 - (1 - 1/50)
B > 49 - 1 + 1/50
B > 48 + 1/50 > 48 (2)
Từ (1) và (2) => 48 < M < 49
=> M không phải số nguyên ( đpcm)
S
12 tháng 3 2017
\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{899}{900}\)
\(=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}....\frac{29.31}{30.30}\)
\(=\frac{1.2.3....29}{2.3.4....30}.\frac{3.4.5....31}{2.3.4....30}\)
\(=\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)
PT
4
CX
3 tháng 5 2016
A=3/4.8/9 .15/16.....899/900
A=1.3/2^2 . 2.4 /3^2 . 3.5/4^2 ....... 29.31 / 30^2
A= 1.2.3.....29 / 2.3.4....30 . 3.4.5...31 / 2.3.4....30
A=1/30 . 31/2
A= 31/60
Nhớ k nha
DM
4
7 tháng 6 2015
\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.......\frac{899}{900}=\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}......\frac{29.31}{30.30}=\frac{1.2.3.....29}{2.3.4......30}.\frac{3.4.5......31}{2.3.4......30}\)
\(=\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)
B
23 tháng 2 2017
Ta có: A=\(\frac{3}{4}.\frac{8}{9}.\frac{15}{16}.\frac{24}{25}....\frac{899}{900}\)
\(\Leftrightarrow A=\frac{1.3}{2^2}.\frac{2.4}{3^2}.\frac{3.5}{4^2}.\frac{4.6}{5^2}....\frac{29.31}{30^2}\)
\(\Leftrightarrow A=\frac{1.2.3.4...29}{2.3.4.5...30}.\frac{3.4.5.6...31}{2.3.4.5...30}\)
\(\Leftrightarrow A=\frac{1}{30}.\frac{31}{2}\)
\(\Leftrightarrow A=\frac{1.31}{30.2}\)
\(\Leftrightarrow A=\frac{31}{60}\)
18 tháng 2 2017
\(A=\frac{3}{4}\times\frac{8}{9}\times\frac{15}{16}\times\frac{24}{25}\times...\times\frac{899}{900}\)
\(=\frac{1.3}{2.2}\times\frac{2.4}{3.3}\times\frac{3.5}{4.4}\times...\times\frac{29.31}{30.30}\)
\(=\frac{\left(1\times2\times3\times...\times29\right)\left(3\times4\times5\times...\times31\right)}{\left(2\times3\times4\times...\times30\right)\left(2\times3\times4\times...\times30\right)}\)
\(=\frac{1\times2\times3\times...\times29}{2\times3\times4\times...\times30}.\frac{3\times4\times5\times...\times31}{2\times3\times4\times...\times30}\)
\(=\frac{1}{30}.\frac{31}{2}\)
\(=\frac{31}{60}\)
NT
19 tháng 2 2017
\(A=\frac{3}{4}.\frac{8}{9}.\frac{15}{16}....\frac{899}{900}\\ =\frac{1.3}{2.2}.\frac{2.4}{3.3}.\frac{3.5}{4.4}....\frac{29.31}{30.30}\\ =\frac{1.2.3.4....29}{2.3.4...30}.\frac{3.4.5...31}{2.3.4...30}\\ =\frac{1}{30}.\frac{31}{2}=\frac{31}{60}\)
.
A =\(\frac{2^2-1}{2^2}\)+ \(\frac{3^2-1}{3^2}\)+ \(\frac{4^2-1}{4^2}\)+,,,+
= 1 - \(\frac{1}{2^2}\)+ 1 - \(\frac{1}{3^2}\)+ ...+ 1 - \(\frac{1}{30^2}\)
= ( 1+ 1+1 +... + 1 ) - ( \(\frac{1}{2^2}\)+ \(\frac{1}{3^2}\)+ ... +\(\frac{1}{30^2}\))
= 29 - ( \(\frac{1}{2^2}\)+ \(\frac{1}{3^2}\)+ ... +\(\frac{1}{30^2}\))
Vậy A không là số nguyên