\(E=-\dfrac{1}{3}\cdot\left(1+2+3\right)-\dfrac{1}{4}\left(1+2+3+4\right)-...-\dfrac{1}{50}\left(1+2+3+...+50\right)\)

\(=\dfrac{-1}{3}\cdot\dfrac{3\cdot4}{2}-\dfrac{1}{4}\cdot\dfrac{4\cdot5}{2}-...-\dfrac{1}{50}\cdot\dfrac{50\cdot51}{2}\)

\(=\dfrac{-4}{2}-\dfrac{5}{2}-...-\dfrac{51}{2}\)

\(=\dfrac{-\left(4+5+...+51\right)}{2}\)

\(=\dfrac{-\left(51+4\right)\cdot\dfrac{48}{2}}{2}=-\dfrac{1320}{2}=-660\)

mình chỉ biết câu b thôi:
 

Ta biến đổi vế phải :

1-1/2+1/3-1/4+.....+1/49-1/50

=(1+1/3+1/5+....+1/49)-(1/2+1/4+1/6+.......+1/50)

=(1+1/2+1/3+.....+1/49+1/50)-2(1/2+1/4+1/6+......+1/50)

=(1+1/2+...+1/50)-(1+1/2+1/3+....+1/25)

=1/26+1/27+.......+1/50

Vậy 1/26+1/27+1/28+.....+1/50=1-1/2+1/3-1/4+......+1/49-1/50

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+\frac{1}{4.5}+...+\frac{1}{49.50}\)

\(1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+\frac{1}{4}-\frac{1}{5}+...+\frac{1}{49}-\frac{1}{50}\)

\(1-\frac{1}{50}\)

\(\frac{49}{50}\)

\(\frac{1}{1.2}+\frac{1}{2.3}+\frac{1}{3.4}+...+\frac{1}{49.50}\)

\(=\frac{1}{1}-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+\frac{1}{3}-\frac{1}{4}+...+\frac{1}{49}-\frac{1}{50}\)

\(=1-\frac{1}{50}\)

\(=\frac{49}{50}\)

\(E=-\frac13\left(1+2+3\right)-\frac14\left(1+2+3+4\right)-\cdots-\frac{1}{50}\left(1+2+3+\cdots+50\right)\)

\(=-\frac13\cdot\frac{3\cdot4}{2}-\frac14\cdot\frac{4\cdot5}{2}-\cdots-\frac{1}{50}\cdot\frac{50\cdot51}{2}\)

\(=-\frac42-\frac52-\cdots-\frac{51}{2}=-\frac12\left(4+5+\cdots+51\right)\)

\(=-\frac12\cdot\frac{\left(51-4+1\right)\left(51+4\right)}{2}=-\frac12\cdot\frac{48\cdot55}{2}=-\frac12\cdot24\cdot55=-12\cdot55=-660\)

a) Số số hạng là :

 ( 50 - 1 ) : 1 + 1 = 50 ( số )

Tổng là :

( 50 + 1 ) x 50 : 2 = 1275 

            Đáp số : 1275

b) không hiểu

đáp số: 1275