bằng 15 hay sao ý

Gọi phân số đã cho là \(A\).

Ta có:

\(A=\dfrac{1\times2015+2\times2014+3\times2013+\cdots+2015\times1}{1\times2+2\times3+3\times4+\cdots+2015\times2016}\)

Tử số:

Ta nhận thấy:

\(1\times2015+2\times2014+\cdots+2015\times1\)

\(=1(2016-1)+2(2016-2)+\cdots+2015(2016-2015)\)

\(=2016(1+2+\cdots+2015)-(1^2+2^2+\cdots+2015^2)\).

Mẫu số:

\(1\times2+2\times3+\cdots+2015\times2016\)

\(=1(1+1)+2(2+1)+\cdots+2015(2015+1)\)

\(=(1^2+2^2+\cdots+2015^2)+(1+2+\cdots+2015)\).

Đặt \(S=1+2+\cdots+2015\), \(T=1^2+2^2+\cdots+2015^2\).

Khi đó:

\(A=\dfrac{2016S-T}{T+S}\).

Với \(S=\dfrac{2015\times2016}{2}\),

\(T=\dfrac{2015\times2016\times4031}{6}\).

Thay vào:

\(A=\dfrac{2016\times\dfrac{2015\times2016}{2}-\dfrac{2015\times2016\times4031}{6}}{\dfrac{2015\times2016\times4031}{6}+\dfrac{2015\times2016}{2}}\)

Rút gọn \(2015\times2016\):

\(A=\dfrac{2016/2-4031/6}{4031/6+1/2}\)

\(=\dfrac{6048-4031}{4031+3}\)

\(=\dfrac{2017}{4034}=\dfrac12\).

=1/2 x 2/3 x 3/4 x 4/5 x 5/6 x.....x2013/2014 - 2014/2015

=1/2014 - 2014/2015

=1/2015

A= 1/2 x 2/3 x 3/4 x ... x 2013/2014 x 2014/2015

A=1/2015

\(\frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{2014}{2015}\)

Khử hết các số giống nhau trên tử và mẫu, ta còn: \(\frac{1}{2015}\)

\(\frac{12}{15}\times\frac{22}{47}+\frac{25}{47}:\frac{15}{12}\)

\(\frac{12}{15}\times\frac{22}{47}+\frac{25}{47}\times\frac{12}{15}\)

\(=\frac{12}{15}\times\left(\frac{22}{47}+\frac{25}{47}\right)\)

\(=\frac{12}{15}\times\frac{22+25}{47}\)

\(=\frac{12}{15}\times1=\frac{12}{15}\)

Giúp mình câu này nhé 11 phần 13 nhân 1/11 x 11/13 x 1/11 + 11/13

a.\(\left(1-\frac{1}{2}\right)\left(1-\frac{1}{3}\right)...\left(1-\frac{1}{2015}\right)=\frac{1}{2}.\frac{2}{3}...\frac{2014}{2015}=\frac{1.2.3...2014}{2.3...2015}=\frac{1}{2015}\)

b.\(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{256}=1-\frac{1}{2}+\frac{1}{2}-\frac{1}{4}+\frac{1}{4}-\frac{1}{8}+...+\frac{1}{128}-\frac{1}{256}=1-\frac{1}{256}=\frac{255}{256}\)

c.\(\frac{5}{2}+\frac{5}{4}+\frac{5}{8}+...+\frac{5}{256}=5\left(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+...+\frac{1}{256}\right)=5.\frac{255}{256}=\frac{1275}{256}\)

d.14,35+(13,7-13,6).1=14,35+0,1.1=14,35+0,1=14,45

a.1/2015