M=\(\dfrac{1}{2}+\left(\dfrac{1}{2}\right)^2+\left(\dfrac{1}{2}\right)^3+.....+\left(\dfrac{1}{2}\right)^{100}\)
ta có:
2M=\(1+\dfrac{1}{2}+\dfrac{1}{2^2}+...+\dfrac{1}{2^{99}}\)
=>2M-M=1-\(\dfrac{1}{2^{100}}\)
=>M=1-\(\dfrac{1}{2^{100}}\)<1(đpcm)
chúc bạn học tốt ^ ^
Tính:
\(M=1+\dfrac{1}{2}\left(1+2\right)+\dfrac{1}{3}\left(1+2+3\right)+...+\dfrac{1}{100}\left(1+2+...+100\right)\)
\(\dfrac{\left(13\dfrac{1}{4}-2\dfrac{5}{27}-10\dfrac{5}{6}\right).230\dfrac{1}{25}+46\dfrac{3}{4}}{\left(1\dfrac{3}{10}+\dfrac{10}{3}\right):\left(12\dfrac{1}{3}-14\dfrac{2}{7}\right)}\)
\(\dfrac{\left(1+2+3+...+99+100\right)\left(\dfrac{1}{2}-\dfrac{1}{3}-\dfrac{1}{7}-\dfrac{1}{9}\right)\left(63.1,2-21.3,6\right)}{1-2+3-4+.....+99-100}\)
Cho A=\(\left(\dfrac{1}{2}-1\right).\left(\dfrac{1}{3}-1\right).\left(\dfrac{1}{4}-1\right)...\left(\dfrac{1}{2015}-1\right).\left(\dfrac{1}{2016}-1\right).\left(\dfrac{1}{2017}-1\right)\)
B=\(\left(-1\dfrac{1}{2}\right).\left(-1\dfrac{1}{3}\right).\left(-1\dfrac{1}{4}\right)...\left(-1\dfrac{1}{2015}\right).\left(-1\dfrac{1}{2016}\right).\left(-1\dfrac{1}{2017}\right)\)
Tính M=A.B
tính: \(\left(\dfrac{1}{2^2}-1\right)\left(\dfrac{1}{3^2}-1\right)\left(\dfrac{1}{4^2-1}\right)...\left(\dfrac{1}{100^2}-1\right)\)
1) Tính
\(A=\dfrac{1}{13}+\dfrac{3}{13.23}+\dfrac{3}{23.33}+...+\dfrac{3}{2003.2013}\)
\(B=\left(\dfrac{1}{2}-1\right).\left(\dfrac{1}{3}-1\right).\left(\dfrac{1}{4}-1\right)....\left(\dfrac{1}{2018}-1\right)\)
2) Tìm x biết:
a) \(x^2-2x-15=0\)
b) \(\dfrac{3}{\left(x+2\right).\left(x+5\right)}+\dfrac{5}{\left(x+5\right).\left(x+10\right)}+\dfrac{7}{\left(x+10\right).\left(x+17\right)}=\dfrac{x+1}{\left(x+2\right).\left(x+17\right)}\)
3) Cho \(\dfrac{a}{b}=\dfrac{d}{c}\) . Chứng minh: \(\dfrac{a^2+d^2}{b^2+c^2}=\dfrac{ad}{bc}\)
4) Cho \(f\left(x\right)=x^{100}-x^{99}+...+x^2-x+1\)
\(g\left(x\right)=-x^{101}+x^{100}-x^{99}+...+x^2-x+1\)
Tính giá trị của hiệu \(f\left(x\right)-g\left(x\right)\) tại x=0,1
5) Cho tam giác ABC có \(\widehat{A}=\ge90\) ; \(M\in AB,N\in AC\)
Chứng minh: BC > MN
6) Cho tam giác ABC, M là trung điểm BC, biết \(\widehat{BAM}>\widehat{CAM}\) . So sánh B và C
A=\(\left(\dfrac{1}{2}-1\right)\left(\dfrac{1}{3}-1\right)\left(\dfrac{1}{4}-1\right)...\left(\dfrac{1}{2017}-1\right)\)
B=\(\left(-1\dfrac{1}{2}\right)\left(-1\dfrac{1}{3}\right)\left(-1\dfrac{1}{4}\right)...\left(-1\dfrac{1}{2017}\right)\)
Tính M=A*B
Tinh
\(\dfrac{\left(1+2+3+...+100\right)\left(\dfrac{1}{3}-\dfrac{1}{5}-\dfrac{1}{7}-\dfrac{1}{9}\right)\left(6,3.12-21.36\right)}{\dfrac{1}{2}+\dfrac{1}{3}+\dfrac{1}{4}+...+\dfrac{1}{100}}\)
\(\dfrac{\left(\dfrac{3}{10}-\dfrac{4}{15}-\dfrac{7}{20}\right).\dfrac{5}{9}}{\left(\dfrac{1}{14}+\dfrac{1}{7}-\dfrac{-3}{35}\right)\dfrac{-4}{3}}\)
Cho B = \(1+\dfrac{1}{2}\left(1+2\right)+\dfrac{1}{3}\left(1+2+3\right)+\dfrac{1}{4}\left(1+2+3+4\right)+...+\dfrac{1}{x}\left(1+2+3+...+x\right)\)
Tìm số nguyên dương để B = 115
Cho \(B=1+\dfrac{1}{2}\left(1+2\right)+\dfrac{1}{3}\left(1+2+3\right)+\dfrac{1}{4}\left(1+2+3+4\right)+...+\dfrac{1}{x}\left(1+2+3+...+x\right)\)
Tìm số nguyên dương x biết B = 115
