\(A=1.2+2.3+3.4+4.5+...+59.60\)

\(\Rightarrow3A=1.2.3+2.3.\left(4-1\right)+...+59.60.\left(61-58\right)\)

\(\Rightarrow3A=1.2.3+2.3.4-1.2.3+...+59.60.61-58.59.60\)

\(\Rightarrow3A=59.60.61\)

\(\Rightarrow A=\frac{59.60.61}{3}\)

3A = 3.4.3 + 4.5.3 + 5.6.3 + ... + 59 . 60 . 3

3A  = 3.4.(5 - 2) + (4.5.(6-3) + 5.6.(7-4) +...+59.60.(61 - 58)  

3A   = 3.4.5 - 2.3.4 + 4.5.6 - 3.4.5 + 5.6.7 - 4.5.6 + ... + 59.60.61 - 58.59.60

3A   = 59.60.61 - 2.3.4

3A   = 215940 - 24

3A   = 215916

  A   = 215916 : 3

  A   = 71972 

\(3A=3.4.3+4.5.3+5.6.3+...+59.60.3\)

\(3A=3.4\left(5-2\right)+4.5\left(6-3\right)+5.6.\left(7-4\right)+...+59.60\left(61-58\right)\)

\(3A=3.4.5-2.3.4+4.5.6-3.4.5+...+59.60.61-58.59.60\)

\(3A=59.60.61-2.3.4\)

\(\Rightarrow A=59.20.61-2.4=...\)

-19,35

\(=-3\left(\dfrac{1}{2}-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}\right)=-3\cdot\dfrac{5}{14}=-\dfrac{15}{14}\)

= 3.(3/3 - 3/4 + 3/4 - 3/5 + 3/5 - 3/6 +.....+ 3/277 - 3/278 + 3/278 - 3/279)

= 3.(3/3 - 3/279 )

= 3.92/93

= 187/93

Mình cũg không chắc chắn là 100% đâu bạn nên dò lại nhé

Đặt biểu thức là \(A\)

\(A=\dfrac{3}{3.4}+\dfrac{3}{4.5}+\dfrac{3}{5.6}+...+\dfrac{3}{278.279}\)

\(\Leftrightarrow\dfrac{1}{3}A=\dfrac{1}{3}\left(\dfrac{3}{3.4}+\dfrac{3}{4.5}+\dfrac{3}{5.6}+...+\dfrac{3}{278.279}\right)\)

\(\Leftrightarrow\dfrac{1}{3}A=\dfrac{1}{3.4}+\dfrac{1}{4.5}+\dfrac{1}{5.6}+...+\dfrac{1}{278.279}\)

\(\Leftrightarrow\dfrac{1}{3}A=\dfrac{1}{3}-\dfrac{1}{4}+\dfrac{1}{4}-\dfrac{1}{5}+...+\dfrac{1}{278}-\dfrac{1}{279}\)

\(\Leftrightarrow\dfrac{1}{3}A=\dfrac{1}{3}-\dfrac{1}{279}\)

\(\Leftrightarrow\dfrac{1}{3}A=\dfrac{93}{279}-\dfrac{1}{279}\)

\(\Leftrightarrow\dfrac{1}{3}A=\dfrac{92}{279}\)

\(\Leftrightarrow A=\dfrac{92}{279}:\dfrac{1}{3}\)

\(\Leftrightarrow A=\dfrac{92}{279}.3\)

\(\Leftrightarrow A=\dfrac{92}{93}\)

\(P=\left(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{59.60}\right).31.32.33....59.60\)

\(\text{Ta có:}\)

\(91=13.7\)

\(\rightarrow4.13+5.17=42.35⋮91\)

\(\left(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{59.60}\right).31.32.33....59.60\)

\(\rightarrow\left(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{59.60}\right).31.32.....60.42.35\)

\(\rightarrow\left(\frac{1}{1.2}+\frac{1}{3.4}+\frac{1}{5.6}+...+\frac{1}{59.60}\right).31.32....60.20.91⋮91\)

Bài này là cơ bản luôn đó:
= 3.(1/1.2 + 1/2.3+...)
= 3.(1/1-1/2+1/2-1/3...)
(tự viết nốt và tính)