CMR : \(\frac{a}{n\left(n+a\right)}=\frac{1}{n}-\frac{1}{n+a}\left(n,a\in N^{\cdot}\right)\)
CMR
\(\frac{a}{n.\left(n+a\right)}=\frac{1}{n}-\frac{1}{n+a}\)(n.a thuộc N*)
1)CMR:
a) \(\frac{1.3.5...39}{21.22.23...40}=\frac{1}{2^{20}}\)
b) \(\frac{1.3.5...\left(2n-1\right)}{\left(n+1\right).\left(n+2\right)\left(n+3\right)...2n}=\frac{1}{2^n}\)( n thuộc N* )
Giải bằng phương pháp quy nạp
CMR với mọi n thuộc N* ta có:
\(a,1.2+2.3+...+n\left(n+1\right)=\frac{n\left(n+1\right)\left(n+2\right)}{3}\)
\(b,\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{2^n}=\frac{2^n-1}{2^n}\)
\(c,1^3+2^3+3^3+...+n^3=\frac{n^2.\left(n+1\right)^2}{4}\)
1) Tính:\(A=3-\frac{1}{2}-\frac{1}{6}-\frac{1}{12}-\frac{1}{20}-\frac{1}{30}-\frac{1}{42}-\frac{1}{56}\)
2) Tìm tất cả các số nguyên tố x,y sao cho x2 - 6y2 - 1 = 0
3) Cho \(n\in N\)biết n-10; n+4. n+60 đều là số nguyên tố. CMR: n+90 là số nguyên tố
4) Tính nhanh
\(A=\left(\frac{7}{9}+1\right)\left(\frac{7}{20}+1\right)\left(\frac{7}{33}+1\right).....\left(\frac{7}{10800}+1\right)\)
Các bn giúp mk nhanh lên nhé
a)Tìm số nguyên dương n thỏa mãn:
\(\frac{1}{2}.\left(1+\frac{1}{1.3}\right).\left(1+\frac{1}{2.4}\right).\left(1+\frac{1}{3.5}\right)...\left(1+\frac{1}{n.\left(n+2\right)}\right)=\frac{2013}{2014}\)
b)tìm a sao cho
\(\left(a+\frac{1}{1.3}\right)+\left(a+\frac{1}{3.5}\right)+\left(a+\frac{1}{5.7}\right)+...+\left(a+\frac{1}{23.25}\right)=11.a+\left(\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}\right)\)
chứng minh : \(\frac{a}{n\times\left(n+a\right)}=\frac{1}{n}-\frac{1}{n+a}\left(n;a\in Nsao\right)\)
CMR \(\forall n\in\)N* ta có
\(\left(1-\frac{1}{2}\right)+\left(\frac{1}{3}-\frac{1}{4}\right)+\left(\frac{1}{5}-\frac{1}{6}\right)+...+\left(\frac{1}{2n-1}-\frac{1}{2n}\right)=\frac{1}{n+1}+\frac{1}{n+2}+...+\frac{1}{2n}\)
Chứng minh rằng :
a, \(\frac{1}{n}-\frac{1}{n+1}=\frac{1}{n\left(n+1\right)}\)
b, \(\frac{1}{n\left(n+q\right)}=\frac{1}{q}\left(\frac{1}{n}-\frac{1}{n+q}\right)\)
