\(\dfrac{1}{a+1}+\dfrac{1}{a\left(a+1\right)}=\dfrac{a\left(a+1\right)}{\left(a+1\right)a\left(a+1\right)}+\dfrac{a+1}{a\left(a+1\right)\left(a+1\right)}\\ =\dfrac{a+1+a\left(a+1\right)}{a\left(a+1\right)\left(a+1\right)}=\dfrac{\left(a+1\right)\left(a+1\right)}{a\left(a+1\right)\left(a+1\right)}\\ =\dfrac{1}{a}\)
Suy ra \(\dfrac{1}{a}=\dfrac{1}{a+1}+\dfrac{1}{a\left(a+1\right)}\)
Chứng minh rằng :
a)\(\dfrac{1}{x}\)-\(\dfrac{1}{x+a}=\dfrac{a}{x\left(x+a\right)}\)
b)\(\dfrac{1}{x\left(x+1\right)}-\dfrac{1}{\left(x+1\right)\left(x+2\right)}=\dfrac{2}{x\left(x+1\right)\left(x+2\right)}\)
c)\(\dfrac{1}{x\left(x+1\right)\left(x+2\right)}-\dfrac{1}{\left(x+1\right)\left(x+2\right)\left(x+3\right)}=\dfrac{3}{x\left(x+1\right)\left(x+2\right)\left(x+3\right)}\)
a) Cho a\(\in\) N. Chứng minh rằng \(\dfrac{1}{a}-\dfrac{1}{a+1}< \dfrac{1}{a^2}< \dfrac{1}{a-1}-\dfrac{1}{a}\)
biết( n,a thuộc n*)
chứng minh: \(\dfrac{a}{n\left(n+a\right)}=\dfrac{1}{n}-\dfrac{1}{n+a}\)
Chứng minh rằng với mọi số tự nhiên khác 0 ta đều có :
a) \(\dfrac{1}{2.5}+\dfrac{1}{5.8}+\dfrac{1}{8.11}+...+\dfrac{1}{\left(3n-1\right).\left(3n+2\right)}=\dfrac{n}{6n+4}\)
b) \(\dfrac{5}{3.7}+\dfrac{5}{7.11}+\dfrac{5}{11.15}+...+\dfrac{5}{\left(4n-1\right).\left(4n+3\right)}=\dfrac{5n}{4n+3}\)
giúp mk với 
Chứng minh: \(\dfrac{1}{2^2}+\dfrac{1}{3^2}+\dfrac{1}{4^2}+...+\dfrac{1}{2015^2}+\dfrac{1}{2015}\)
Cho biểu thức: A=\(\dfrac{2}{n-1}\left(n\in Z\right).\)Tìm tất cả các giá trị nguyên của n để A là số nguyên.
Không quy đồng mẫu hãy so sánh a và B biết; A=\(\dfrac{12}{5^{2012}}+\dfrac{18}{5^{2013}}\); B=\(\dfrac{18}{5^{2012}}+\dfrac{12}{5^{2013}}\)
Cho A\(=\dfrac{\left(3\dfrac{2}{15}+\dfrac{1}{5}\right):2\dfrac{1}{2}}{\left(5\dfrac{3}{7}-2\dfrac{1}{4}\right):4\dfrac{43}{56}}\)
B\(=\dfrac{1,2:\left(1\dfrac{1}{5}\cdot1\dfrac{1}{4}\right)}{0,32+\dfrac{2}{25}}\)
Chứng tỏ A=B
Tìm x \(\in\) Z :
a \(\left(\dfrac{3}{4}\right)^{2x}=\dfrac{12.5^9.\left(-7\right)^{10}+3.7^9.\left(-5\right)^{11}}{35^9.2^4}\)
b, \(\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+....+\dfrac{2}{x\left(x+1\right)}=\dfrac{499}{1000}\)
NHANH NHÁ GIÚP MINK VỚI GIẢI RÕ CHI TIẾT MAI MIK NỘP RỒI ! THANKS TRC^^
Cho a1,a2,..., a10 \(\in\) N* và 1<a1<a2<...<a10
\(CMR:\left(1+\dfrac{1}{a_1^2}\right)\times\left(1+\dfrac{1}{a_2^2}\right)\times...\times\left(1+\dfrac{1}{a_{10}^2}\right)< 2\)
1)tìm x\(\in\)Q biết:
a)\(\left(5\cdot x+1\right)^2=\dfrac{36}{49}\)
2)tìm x,y biết:
a)\(x^2+\left(y-\dfrac{1}{10}\right)^4=0\)
b)\(\left(\dfrac{1}{2}x-5\right)^{20}+\left(y^2-\dfrac{1}{4}\right)^{40}\le0\)
