tính
\(C=\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+.........+\frac{1}{2499}\)
tính nhanh:
C=\(\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+........+\frac{1}{2499}\)
\(C=\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+.....+\frac{1}{49.51}\)
\(C=\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{49}-\frac{1}{51}\right)\)
\(C=\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{51}\right)\)
\(C=\frac{1}{2}.\frac{16}{51}=\frac{8}{51}\)
\(C=\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{5}+...+\frac{1}{49}-\frac{1}{51}\right)\)
\(C=\frac{1}{2}.\left(\frac{1}{3}-\frac{1}{51}\right)=\frac{8}{51}\)
C=\(\frac{1}{3.5}\)+\(\frac{1}{5.7}\)+\(\frac{1}{7.9}\)+......+\(\frac{1}{49.51}\)
C=(\(\frac{2}{3.5}\)+\(\frac{2}{5.7}\)+\(\frac{2}{7.9}\)+.......+\(\frac{2}{49.51}\)) :2
C=(\(\frac{1}{3}\)-\(\frac{1}{5}\)+\(\frac{1}{5}\)-\(\frac{1}{7}\)+\(\frac{1}{7}\)-\(\frac{1}{9}\)+....+\(\frac{1}{49}\)-\(\frac{1}{51}\)) :2
C=(\(\frac{1}{3}\)-\(\frac{1}{51}\)) :2
C=\(\frac{16}{51}\):2=\(\frac{8}{51}\)
\(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}=?\)
Ta có :
\(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}\)
\(=\)\(\frac{1}{2}\left(\frac{2}{3}+\frac{2}{15}+\frac{2}{35}+...+\frac{2}{2499}\right)\) ( bước này hơi khó hiểu tí nhé )
\(=\)\(\frac{1}{2}\left(\frac{2}{1.3}+\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{49.51}\right)\) ( phân tích mẫu )
\(=\)\(\frac{1}{2}\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{49}-\frac{1}{51}\right)\) ( áp dụng công thức thoi )
\(=\)\(\frac{1}{2}\left(1-\frac{1}{51}\right)\) ( loại bỏ nhưng phân số đối nhau )
\(=\)\(\frac{1}{2}.\frac{50}{51}\)
\(=\)\(\frac{25}{51}\)
Chúc bạn học tốt ~
ĐẶT \(A\)\(=\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+...+\frac{1}{49\cdot51}\)
\(2.A=\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+...+\frac{2}{49\cdot51}\)
\(2.A=\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{49}-\frac{1}{51}\)
\(2.A=1-\frac{1}{51}\)
\(2.A=\frac{50}{51}\)
\(\Rightarrow A=\frac{50}{51}\div2=\frac{25}{51}\)
Chứng Minh Rằng:
C = \(\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}\)
D = \(\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{10^2}< 1\)
Câu \(C=\left(...\right)\) thiếu đề
Ta có :
\(\frac{1}{2^2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}< \frac{1}{2.3}\)
\(............\)
\(\frac{1}{10^2}< \frac{1}{9.10}\)
\(\Rightarrow\)\(D=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{10^2}< \frac{1}{1.2}+\frac{1}{2.3}+...+\frac{1}{9.10}\)
\(\Rightarrow\)\(D< \frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{9}-\frac{1}{10}\)
\(\Rightarrow\)\(D< 1-\frac{1}{10}< 1\)
\(\Rightarrow\)\(D< 1\) ( đpcm )
Vậy \(D< 1\)
Chúc bạn học tốt ~
\(C=\frac{1}{15}+\frac{1}{35}+....+\frac{1}{2499}\)
\(C=\frac{1}{3.5}+\frac{1}{5.7}+........+\frac{1}{49.51}\)
\(C=\frac{1}{2}\left(\frac{2}{3.5}+\frac{2}{5.7}+.....+\frac{2}{49.51}\right)\)
\(C=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+......+\frac{1}{49}-\frac{1}{51}\right)\)
\(C=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{51}\right)\)
\(C=\frac{1}{2}.\frac{16}{51}\)
\(C=\frac{8}{51}\)
\(D=\frac{1}{2^2}+\frac{1}{3^2}+......+\frac{1}{10^2}\)
ta có :
\(\frac{1}{2^2}=\frac{1}{2.2}< \frac{1}{1.2}\)
\(\frac{1}{3^2}=\frac{1}{3.3}< \frac{1}{2.3}\)
\(....................\)
\(\frac{1}{10^2}=\frac{1}{10.10}< \frac{1}{9.10}\)
\(\Rightarrow\frac{1}{2^2}+\frac{1}{3^2}+......+\frac{1}{10^2}< \frac{1}{1.2}+\frac{1}{2.3}+........+\frac{1}{9.10}\)
\(\Rightarrow D< 1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+........+\frac{1}{9}-\frac{1}{10}\)
\(\Rightarrow D< 1-\frac{1}{10}\)
\(\Rightarrow D< \frac{9}{10}\) ( 1 )
mà \(\frac{9}{10}< 1\) ( 2 )
từ ( 1 ) và ( 2 ) \(\Rightarrow D< 1\left(\text{đ}pcm\right)\)
C = \(\frac{1}{15}\) + \(\frac{1}{35}\)+ ...... + \(\frac{1}{2499}\)
Trả lời
\(C=\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}\)
\(\Rightarrow C=\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+\frac{...1}{49\cdot51}\)
\(\Rightarrow2C=2\left(\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+...+\frac{1}{49\cdot51}\right)\)
\(\Rightarrow2C=\frac{2}{1\cdot3}+\frac{2}{5\cdot7}+...+\frac{2}{49\cdot51}\)
\(\Rightarrow2C=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{49}-\frac{1}{51}\)
\(\Rightarrow2C=1-\frac{1}{51}\)
\(\Rightarrow2C=\frac{50}{51}\)
\(\Rightarrow C=\frac{50}{51}:2\)
\(\Rightarrow C=\frac{25}{51}\)
Vậy \(C=\frac{25}{51}\)
\(C=\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}\)
\(=\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{49.51}\)
\(=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{49}-\frac{1}{50}\right)\)
\(=\frac{1}{2}\left(\frac{1}{3}-\frac{1}{50}\right)\)\(=\frac{1}{2}.\frac{47}{150}\)
\(=\frac{47}{300}\)\(\Rightarrow C=\frac{47}{300}\)
Tính nhanh
\(1\frac{7}{15}-\frac{1}{3}-\frac{1}{15}-\frac{1}{35}-\frac{1}{63}-\frac{1}{99}-\frac{1}{143}-\frac{1}{195}\)
Đặt \(A=1\frac{7}{15}-\frac{1}{3}-\frac{1}{15}-\frac{1}{35}-\frac{1}{63}-\frac{1}{99}-\frac{1}{143}-\frac{1}{195}\)
\(\Rightarrow A=\frac{22}{15}-\left(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}+\frac{1}{195}\right)\)
Đặt \(B=\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+\frac{1}{99}+\frac{1}{143}+\frac{1}{195}\)
\(\Rightarrow B=\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+\frac{1}{7\cdot9}+\frac{1}{9\cdot11}+\frac{1}{11\cdot13}+\frac{1}{13\cdot15}\)
\(\Rightarrow2B=2\left(\frac{1}{1\cdot3}+\frac{1}{3\cdot5}+\frac{1}{5\cdot7}+\frac{1}{7\cdot9}+\frac{1}{9\cdot11}+\frac{1}{11\cdot13}+\frac{1}{13\cdot15}\right)\)
\(\Rightarrow2B=\frac{2}{1\cdot3}+\frac{2}{3\cdot5}+\frac{2}{5\cdot7}+\frac{2}{7\cdot9}+\frac{2}{9\cdot11}+\frac{2}{11\cdot13}+\frac{2}{13\cdot15}\)
\(\Rightarrow2B=1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+\frac{1}{9}-\frac{1}{11}+\frac{1}{11}-\frac{1}{13}+\frac{1}{13}-\frac{1}{15}\)
\(\Rightarrow2B=1-\frac{1}{15}\)
\(\Rightarrow2B=\frac{14}{15}\)
\(\Rightarrow B=\frac{14}{15}:2\Rightarrow B=\frac{7}{15}\)
\(\Rightarrow A=\frac{22}{15}-\frac{7}{15}\Rightarrow A=\frac{15}{15}=1\)
=22/15- 1/1.3 - 1/3.5 - 1/5.7 -.........- 1/11.13 - 1/13.15
=22/15 - (1/1.3+1/3.5+....+1/13.17)
=22/15 - 1/2(2/1.3+2/3.5.........+2/13.17)
=22/15 - 1/2(1-1/3+1/3-1/4+.............+1/13-1/17)
=22/15 - 1/2(1-1/17)
=22/15-8/17
=254/255
Tính:
a) \(\frac{5}{1.4}+\frac{5}{4.7}+...+\frac{5}{100.103}\)
b) \(\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}\)
a)\(\frac{5}{1.4}+\frac{5}{4.7}+...+\frac{5}{100.103}=\frac{5}{3}\cdot\left(\frac{3}{1.4}+\frac{4}{4.7}+...+\frac{3}{100.103}\right)\)
\(=\frac{5}{3}\cdot\left(1-\frac{1}{4}+\frac{1}{4}-\frac{1}{7}+...+\frac{1}{100}-\frac{1}{103}\right)=\frac{5}{3}\cdot\left(1-\frac{1}{103}\right)=\frac{5}{3}\cdot\frac{102}{103}=\frac{170}{103}\)b)\(\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}=\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{49.51}\)
\(\frac{1}{2}\cdot\left(\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{49}-\frac{1}{51}\right)=\frac{1}{2}\cdot\left(\frac{1}{3}-\frac{1}{51}\right)=\frac{1}{2}\cdot\frac{16}{51}=\frac{8}{51}\)
Câu a) bạn Ác Mộng làm rồi nên mình làm b) nha
b)Gọi A = \(\frac{1}{15}+\frac{1}{35}+...+\frac{1}{2499}\)
\(A=\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{49.51}\)
\(2A=2.\left(\frac{1}{3.5}+\frac{1}{5.7}+...+\frac{1}{49.51}\right)\)
\(2A=\frac{2}{3.5}+\frac{2}{5.7}+...+\frac{2}{49.51}\)
\(2A=\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+...+\frac{1}{49}-\frac{1}{51}\)
\(2A=\frac{1}{3}-\frac{1}{51}\)
\(2A=\frac{16}{51}\)
\(A=\frac{16}{51}:2\)
\(A=\frac{8}{51}\)
5*(5-5/4+5/4-5/7+.......+5/100-5/103)
5*(5-5/103)
5*......... bạn tự tính nhé
câu b 1/3*5+1/5*7+............+1/49*51
1*(1/1-1/3+1/3-1/5+............+1/49-1/51)
1/1-1/51 tính ra rồi ra kết quả
tk nha
\(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+...+\frac{1}{195}\)
Tính nhanh
\(S=\frac{1}{1.3}+\frac{1}{3.5}+...+\frac{1}{13.15}\)
\(\Rightarrow S=\frac{1}{2}.\left(\frac{1}{1}-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+...+\frac{1}{13}-\frac{1}{15}\right)\)
\(\Rightarrow S=\frac{1}{2}.\left(1-\frac{1}{15}\right)\)
\(\Rightarrow S=\frac{1}{2}.\frac{14}{15}\)
\(\Rightarrow S=\frac{7}{15}\)
\(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+....+\frac{1}{195}\)
\(=\frac{1}{1x3}+\frac{1}{3x5}+\frac{1}{5x7}+\frac{1}{7x9}+...+\frac{1}{13x15}\)
\(=\frac{1}{2}x\left(\frac{2}{1x3}+\frac{2}{3x5}+\frac{2}{5x7}+\frac{2}{7x9}+...+\frac{2}{13x15}\right)\)
\(=\frac{1}{2}x\left(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{13}-\frac{1}{15}\right)\)
\(=\frac{1}{2}x\left(1-\frac{1}{15}\right)=\frac{1}{2}x\frac{14}{15}=\frac{7}{15}\)
\(\frac{1}{3}\)+\(\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+...+\)\(\frac{1}{195}\)
=\(\frac{1}{1.3}+\frac{1}{3.5}+\frac{1}{5.7}+\frac{1}{7.9}+...+\frac{1}{13.15}\)
=(\(1-\frac{1}{3}+\frac{1}{3}-\frac{1}{5}+\frac{1}{5}-\frac{1}{7}+\frac{1}{7}-\frac{1}{9}+...+\frac{1}{13}-\frac{1}{15}\)) :2
=\(\left(1-\frac{1}{15}\right)\):2
=\(\frac{14}{15}:2\)
=\(\frac{7}{15}\)
mk giải đúng ko vậy , sai thì nói ra mk còn sửa nha
Tính nhanh hợp lí:
a)\(A=\frac{1}{6}+\frac{1}{12}+\frac{1}{20}+...+\frac{1}{600}\)
b)\(B=\frac{-5}{3}+\frac{-5}{15}+\frac{-5}{35}+...+\frac{-5}{2499}\)
tính nhanh:\(\frac{1}{3}+\frac{1}{15}+\frac{1}{35}+\frac{1}{63}+.....+\frac{1}{195}\)
1/3 + 1/15 + 1/35+ 1/63 +...... + 1/195
= 1/3 + 1/3x5 + 1/5 x7 + 1/7x9 + ....+1/13x15
= 1/3+1/3-1/5+1/5-1/7+1/7-1/9+....+1/13-1/15 ( vì +- nên rút gọn )
= 1/3+1/3-1/15
=3/5
=1/1.3+1/3.5+1/5.7+...+1/13.15
=1/2.2(1/1.3+1/3.5+1/5.7+...+1/13.15)
=1/2(2/1.3+2/3.5+2/5.7+...+2/13.15)
=1/2(1-1/3+1/3-1/5+1/5-1/7+...+1/13-1/15)
=1/2[(1-1/15)+(1/3-1/3)+(1/5-1/5)+...+(1/13-1/15)]
=1/2[(1-1/15)+0+...+0=1/2(1-1/15)=1/2.14/15=14/30=7/15
1/3 + 1/15 + 1/35+ 1/63 +...... + 1/195
= 1/3 + 1/3x5 + 1/5 x7 + 1/7x9 + ....+1/13x15
= 1/3+1/3-1/5+1/5-1/7+1/7-1/9+....+1/13-1/15 ( vì +- nên rút gọn )
= 1/3+1/3-1/15
=3/5