quy luật +3+5+7+9+11+........

bai toan nay kho

bài toán này khó duyệt đi

Cái này là tính 1*1+2*2+3*3+4*4+......+50*50

=(1-1/3)(1-1/4)(1-1/5)*...*(1-1/50)(1+1/3)(1+1/4)*...*(1+1/50)

=2/3*3/4*...*49/50*4/3*5/4*...*51/50

=2/50*51/3=17*1/25=17/25

\(\left(1-\dfrac{1}{9}\right)\cdot\left(1-\dfrac{1}{16}\right)\cdot\left(1-\dfrac{1}{25}\right)\cdot...\cdot\left(1-\dfrac{1}{2500}\right)\)

\(=\left(\dfrac{9}{9}-\dfrac{1}{9}\right)\cdot\left(\dfrac{16}{16}-\dfrac{1}{16}\right)\cdot...\cdot\left(\dfrac{2500}{2500}-\dfrac{1}{2500}\right)\)

\(=\dfrac{8}{9}\cdot\dfrac{15}{16}\cdot\dfrac{24}{25}\cdot...\cdot\dfrac{2499}{2500}\)

\(=\dfrac{8\cdot15\cdot24\cdot...\cdot2499}{9\cdot16\cdot25\cdot...\cdot2500}\)

\(=\dfrac{\left(2\cdot4\right)\cdot\left(3\cdot5\right)\cdot\left(4\cdot6\right)\cdot....\cdot\left(49\cdot51\right)}{\left(3\cdot3\right)\cdot\left(4\cdot4\right)\cdot\left(5\cdot5\right)\cdot...\cdot\left(50\cdot50\right)}\)

\(=\dfrac{\left(2\cdot3\cdot4\cdot5\cdot...\cdot49\right)\left(4\cdot5\cdot6\cdot...\cdot51\right)}{\left(2\cdot3\cdot4\cdot...\cdot50\right)\left(2\cdot3\cdot4\cdot...\cdot50\right)}\)

\(=\dfrac{1\cdot51}{50\cdot2}\)

\(=\dfrac{51}{100}\)

\(A=\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{100}\)

\(\Rightarrow A=1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}...+\dfrac{1}{99}-\dfrac{1}{100}\)

\(\Rightarrow A=1-\dfrac{1}{100}\)

\(\Rightarrow A=\dfrac{99}{100}\)

=1/2+1/3+1/4+...+1/100

xét mẫu:có ssh là (100-2):1+1=99 số

tổng là (100+2)*99:2=5940

vậy ta có 1/5940

\(\left(1-\frac{1}{4}\right)\times\left(1-\frac{1}{9}\right)\times\left(1-\frac{1}{16}\right)\times\left(1-\frac{1}{25}\right)\times\left(1-\frac{1}{36}\right)\)

\(=\frac{3}{4}\times\frac{8}{9}\times\frac{15}{16}\times\frac{24}{25}\times\frac{36}{36}\)

\(=\frac{1.3}{2.2}\times\frac{2.4}{3.3}\times\frac{3.5}{4.4}\times\frac{4.6}{5.5}\times\frac{5.7}{6.6}\)

\(=\frac{1.2.3.4.5}{2.3.4.5.6}\times\frac{3.4.5.6.7}{2.3.4.5.6}\)

\(=\frac{1}{6}\times\frac{7}{2}\)

\(=\frac{7}{12}\)

(1-1/4)×(1-1/9)×(1-1/16)×(1-1/25)×(1-1/36)

=(4/4-1/4)×(9/9-1/9)×(16/16-1/16)×(25/25-1/25)×(36/36-1/36)

=3/4×8/9×15/16×24/25×35/36

=1×3×2×4×3×5×4×6×5×7/2×2×3×3×4×4×5×5×6×6

=(1×2×3×4×5)×(3×4×5×6×7)/(2×3×4×5×6)×(2×3×4×5×6)

=1/6×7/2

=7/12

Thank you

OK chờ tí nhé

câu a tách ra nha

câu b thì tính trong ngoặc rồi tách

hok tốt

...

Câu a tham khảo link này nha bn

https://olm.vn/hoi-dap/detail/186073292336.html

\(N=\frac{1}{2^2}+\frac{1}{3^2}+...+\frac{1}{10^2}\)

\(N>\frac{1}{2.3}+\frac{1}{3.4}+....+\frac{1}{10.11}\)

\(N>\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-....-\frac{1}{11}=\frac{1}{2}-\frac{1}{11}=\frac{10}{22}>\frac{9}{22}\)

Vậy N > 9/22