a: \(\dfrac{1}{2}\cdot2^n+4\cdot2^n=9\cdot5^n\)

=>\(2^n\cdot\left(\dfrac{1}{2}+4\right)=5^n\cdot9\)

=>\(2^n\cdot\dfrac{9}{2}=5^n\cdot9\)

=>\(2^{n-1}=5^n\)

=>\(n-1=n\cdot log_25\)

=>\(n\left(1-log_25\right)=1\)

=>\(n=\dfrac{1}{1-log_25}\)

\(\dfrac{x-10}{30}+\dfrac{x-14}{43}+\dfrac{x-5}{95}+\dfrac{x-148}{8}=0\\ \Rightarrow\left(\dfrac{x-10}{30}-3\right)+\left(\dfrac{x-14}{43}-2\right)+\left(\dfrac{x-5}{95}-1\right)+\left(\dfrac{x-148}{8}+3\right)=0\\ \Rightarrow\dfrac{x-100}{30}+\dfrac{x-100}{43}+\dfrac{x-100}{95}+\dfrac{x-100}{8}=0\\ \Rightarrow\left(x-100\right)\left(\dfrac{1}{30}+\dfrac{1}{43}+\dfrac{1}{95}+\dfrac{1}{8}\right)=0\\ \Rightarrow x-100=0\\ \Rightarrow x=100\)

a)

 \(\dfrac{x+1}{10}+\dfrac{x+1}{11}+\dfrac{x+1}{12}=\dfrac{x+1}{13}+\dfrac{x+1}{14}\\ \Rightarrow\dfrac{x+1}{10}+\dfrac{x+1}{11}+\dfrac{x+1}{12}-\dfrac{x+1}{13}-\dfrac{x+1}{14}=0\\ \Rightarrow\left(x+1\right)\cdot\left(\dfrac{1}{10}+\dfrac{1}{11}+\dfrac{1}{12}-\dfrac{1}{13}-\dfrac{1}{14}\right)=0\)

Vì \(\dfrac{1}{10}+\dfrac{1}{11}+\dfrac{1}{12}-\dfrac{1}{13}-\dfrac{1}{14}\ne0\) nên:

\(x+1=0\\ \Rightarrow x=-1\)

Vậy...

b) \(\dfrac{315-x}{101}+\dfrac{313-x}{103}+\dfrac{311-x}{105}+\dfrac{309-x}{107}=-4\\ \Rightarrow\dfrac{315-x}{101}+\dfrac{313-x}{103}+\dfrac{311-x}{105}+\dfrac{309-x}{107}+4=0\\ \Rightarrow\left(\dfrac{315-x}{101}+1\right)+\left(\dfrac{313-x}{103}+1\right)+\left(\dfrac{311-x}{105}+1\right)+\left(\dfrac{309-x}{107}+1\right)=0\\ \Rightarrow\dfrac{416-x}{101}+\dfrac{416-x}{103}+\dfrac{416-x}{105}+\dfrac{416-x}{107}=0\\ \Rightarrow\left(416-x\right)\left(\dfrac{1}{101}+\dfrac{1}{103}+\dfrac{1}{105}+\dfrac{1}{107}\right)=0\)

Vì \(\dfrac{1}{101}+\dfrac{1}{103}+\dfrac{1}{105}+\dfrac{1}{107}\ne0\) nên:

\(416-x=0\\ \Rightarrow x=416\)

Vậy...

Hiệu số tuổi giữa tuổi cha và tuổi con là:

            43-12=31(tuổi)

Ta có sơ đồ sau khi tuổi cha gấp tuổi con 2 lần 

  Tuổi cha 2 phần

  Tuổi con 1 phần 

  Hiệu là 31t ở 1 phần còn lại

Hiệu số phần bằng nhau là :

   2-1=1 phần

Gía trị 1 phần hay tuổi con lúc tuổi cha gấp đôi tuổi con là

          31:1=31(tuổi)

Sau số năm thì tuổi cha gấp đôi tuổi con là:

        31-12=19(năm)

             Đ/S: 19 năm

Bài 8:

1: \(\dfrac{-20}{41}\cdot\dfrac{-4}{5}=\dfrac{20}{5}\cdot\dfrac{4}{41}=4\cdot\dfrac{4}{41}=\dfrac{16}{41}\)

2: \(\dfrac{-24}{5}\cdot\dfrac{15}{-8}=\dfrac{-24}{-8}\cdot\dfrac{15}{5}=3\cdot3=9\)

3: \(\dfrac{-4}{34}\cdot\dfrac{17}{-24}=\dfrac{4}{24}\cdot\dfrac{17}{34}=\dfrac{1}{2}\cdot\dfrac{1}{6}=\dfrac{1}{12}\)

Bài 6:

1: \(2\dfrac{3}{5}-1\dfrac{2}{3}=\dfrac{13}{5}-\dfrac{5}{3}=\dfrac{39}{15}-\dfrac{25}{15}=\dfrac{14}{15}\)

2: \(3\dfrac{3}{7}+2\dfrac{1}{2}=\dfrac{24}{7}+\dfrac{5}{2}=\dfrac{48}{14}+\dfrac{35}{14}=\dfrac{83}{14}\)

3: \(-3\dfrac{1}{2}-2\dfrac{1}{4}=\dfrac{-7}{2}-\dfrac{9}{4}=\dfrac{-14}{4}-\dfrac{9}{4}=-\dfrac{23}{4}\)

4: \(-2\dfrac{1}{2}-3\dfrac{1}{4}=-\dfrac{5}{2}-\dfrac{13}{4}=-\dfrac{10}{4}-\dfrac{13}{4}=-\dfrac{23}{4}\)

5: \(-4\dfrac{1}{2}+2\dfrac{3}{10}=-\dfrac{9}{2}+\dfrac{23}{10}=-\dfrac{45}{10}+\dfrac{23}{10}=-\dfrac{22}{10}=-\dfrac{11}{5}\)

6: \(-6\dfrac{1}{7}-\left(-7\dfrac{1}{6}\right)=-6-\dfrac{1}{7}+7+\dfrac{1}{6}\)

\(=1-\dfrac{1}{7}+\dfrac{1}{6}=\dfrac{6}{7}+\dfrac{1}{6}=\dfrac{43}{42}\)

Bài 7:

\(1)\dfrac{2}{7}+\dfrac{6}{21}-\dfrac{3}{14}\\ =\dfrac{12}{42}+\dfrac{12}{42}-\dfrac{12}{42}\\ =\dfrac{12}{42}\\ =\dfrac{2}{7}\\ 2)-\dfrac{7}{2}+\dfrac{3}{4}-\dfrac{17}{12}\\ =\dfrac{-42}{12}+\dfrac{9}{12}-\dfrac{17}{12}\\ =\dfrac{-50}{12}=\dfrac{-25}{6}\\ 3)\dfrac{1}{12}+\dfrac{1}{4}+\dfrac{2}{3}\\ =\dfrac{1}{12}+\dfrac{3}{12}+\dfrac{8}{12}=\dfrac{12}{12}=1\\ 4)\dfrac{1}{3}+\dfrac{-4}{5}-\dfrac{8}{15}\\ =\dfrac{5}{15}+\dfrac{-12}{15}-\dfrac{8}{15}=-\dfrac{15}{15}=-1\\ 5)\dfrac{2}{3}+\dfrac{-3}{4}+\dfrac{2}{6}\\ =\dfrac{2}{3}+\dfrac{-3}{4}+\dfrac{1}{3}\\ =\left(\dfrac{2}{3}+\dfrac{1}{3}\right)+\dfrac{-3}{4}\\ =1+\dfrac{-3}{4}\\ =\dfrac{1}{4}\\ 6\text{ })-\dfrac{5}{18}+\dfrac{5}{45}-\dfrac{9}{6}\\ =-\dfrac{5}{18}+\dfrac{1}{9}-\dfrac{3}{2}\\ =-\dfrac{5}{18}+\dfrac{2}{18}-\dfrac{27}{18}\\ =-\dfrac{30}{18}\\ =-\dfrac{5}{3}\)

Bài 8:

1: \(\dfrac{-20}{41}\cdot\dfrac{-4}{5}=\dfrac{20}{5}\cdot\dfrac{4}{41}=4\cdot\dfrac{4}{41}=\dfrac{16}{41}\)

2: \(\dfrac{-24}{5}\cdot\dfrac{15}{-8}=\dfrac{-24}{-8}\cdot\dfrac{15}{5}=3\cdot3=9\)

3: \(\dfrac{-4}{34}\cdot\dfrac{17}{-24}=\dfrac{4}{24}\cdot\dfrac{17}{34}=\dfrac{1}{2}\cdot\dfrac{1}{6}=\dfrac{1}{12}\)

Bài 6:

1: \(2\dfrac{3}{5}-1\dfrac{2}{3}=\dfrac{13}{5}-\dfrac{5}{3}=\dfrac{39}{15}-\dfrac{25}{15}=\dfrac{14}{15}\)

2: \(3\dfrac{3}{7}+2\dfrac{1}{2}=\dfrac{24}{7}+\dfrac{5}{2}=\dfrac{48}{14}+\dfrac{35}{14}=\dfrac{83}{14}\)

3: \(-3\dfrac{1}{2}-2\dfrac{1}{4}=\dfrac{-7}{2}-\dfrac{9}{4}=\dfrac{-14}{4}-\dfrac{9}{4}=-\dfrac{23}{4}\)

4: \(-2\dfrac{1}{2}-3\dfrac{1}{4}=-\dfrac{5}{2}-\dfrac{13}{4}=-\dfrac{10}{4}-\dfrac{13}{4}=-\dfrac{23}{4}\)

5: \(-4\dfrac{1}{2}+2\dfrac{3}{10}=-\dfrac{9}{2}+\dfrac{23}{10}=-\dfrac{45}{10}+\dfrac{23}{10}=-\dfrac{22}{10}=-\dfrac{11}{5}\)

6: \(-6\dfrac{1}{7}-\left(-7\dfrac{1}{6}\right)=-6-\dfrac{1}{7}+7+\dfrac{1}{6}\)

\(=1-\dfrac{1}{7}+\dfrac{1}{6}=\dfrac{6}{7}+\dfrac{1}{6}=\dfrac{43}{42}\)

a: Ta có: ΔOCK vuông tại C

=>\(CK^2+CO^2=OK^2\)

=>\(CK=\sqrt{10^2-8^2}=6\left(cm\right)\)

Xét ΔOCK vuông tại C có CA là đường cao

nên \(OA\cdot OK=OC^2;CA\cdot OK=CO\cdot CK\)

=>\(OA=\dfrac{OC^2}{OK}=\dfrac{8^2}{10}=6,4\left(cm\right);CA=\dfrac{6\cdot8}{10}=4,8\left(cm\right)\)

Xét ΔCOK vuông tại C có \(sinCOK=\dfrac{CK}{OK}=\dfrac{6}{10}=\dfrac{3}{5}\)

nên \(\widehat{COK}=\widehat{xOy}\simeq36^052'\)

b: Xét ΔCAO vuông tại A có AH là đường cao

nên \(CH\cdot CO=CA^2\left(1\right)\)

Xét ΔCOK vuông tại O có CA là đường cao

nên \(AO\cdot AK=AC^2\left(2\right)\)

Từ (1),(2) suy ra \(CH\cdot CO=AO\cdot AK\)

Tớ sẽ làm mẫu cho cậu 1 số bài nhé:

a) \(A=\dfrac{6-5}{5.6}+\dfrac{7-6}{6.7}+...+\dfrac{25-24}{24.25}\)

   \(A=\dfrac{6}{5.6}-\dfrac{5}{5.6}+\dfrac{7}{6.7}-\dfrac{6}{6.7}+...+\dfrac{25}{24.25}-\dfrac{24}{24.25}\)

   \(A=\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+...+\dfrac{1}{24}-\dfrac{1}{25}\)

   \(A=\dfrac{1}{5}-\dfrac{1}{25}\)

   \(A=\dfrac{4}{25}\)

Bài 4; 2 và câu d bài 1 cậu sẽ cần phải đưa tử về hiệu giữa 2 thừa số ở mẫu.

\(\dfrac{4}{5}\) K = \(\dfrac{7-3}{3.7}+\dfrac{11-7}{7.11}+\dfrac{15-11}{11.15}=...+\dfrac{85-81}{81.85}+\dfrac{89-85}{85.89}\)

\(\dfrac{4}{5}K=\dfrac{1}{3}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{11}+\dfrac{1}{11}-\dfrac{1}{15}+...+\dfrac{1}{81}-\dfrac{1}{85}+\dfrac{1}{85}-\dfrac{1}{89}\)

\(\dfrac{4}{5}K=\dfrac{1}{3}-\dfrac{1}{89}\)

\(\dfrac{4}{5}K=\dfrac{43}{147}\)

\(K=\dfrac{43}{147}\div\dfrac{4}{5}\)

\(K=\dfrac{215}{588}\)

Với bài 3 thì cậu chỉ cần đảo vị trí từ dưới lên trên là được nhé.

Bài 5: (Viết lại tổng E). Khoảng cách giữa 2 thừa số ở mẫu là 6, cậu hãy nhân tử với 6, tính sau đó : 6 nhé.

→ E = \(\dfrac{1}{1.7}+\dfrac{1}{7.13}+\dfrac{1}{13.19}+\dfrac{1}{19.25}+\dfrac{1}{25.31}+\dfrac{1}{31.37}\)

\(C=\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dfrac{2}{7.9}+\dfrac{2}{9.11}+\dfrac{2}{11.13}\)

Bài 6. Quan sát:

\(3\left(\dfrac{1}{5}-\dfrac{3}{x-2}\right)=\dfrac{24}{35}\) và tương tự như câu b, luôn là cái đầu tiên - cái cuối cùng.

Bài 7. Cậu trừ 1 ở cả 2 vế rồi nhân \(\dfrac{1}{2}\).

\(\dfrac{1}{6}+\dfrac{1}{12}+\dfrac{1}{20}+...+\dfrac{1}{x\left(x+1\right)}\)

= \(\dfrac{1}{2.3}+\dfrac{1}{3.4}+\dfrac{1}{4.5}+...+\dfrac{1}{x\left(x+1\right)}\)

Cậu cứ làm như những bài trên nhé.

Bài 1: 

\(a,A=\dfrac{1}{5.6}+\dfrac{1}{6.7}+...+\dfrac{1}{24.25}\)

\(=\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+...+\dfrac{1}{24}-\dfrac{1}{25}\)

\(=\dfrac{1}{5}-\dfrac{1}{25}=>\dfrac{5}{25}-\dfrac{1}{25}\)

\(=\dfrac{4}{25}\)

\(b,B=\dfrac{2}{1.3}+\dfrac{2}{3.5}+\dfrac{2}{5.7}+...+\dfrac{2}{99.101}\)

       \(=1.\left(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{99.101}\right)\)

       \(=1.\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\)

       \(=1.\left(1-\dfrac{1}{101}\right)\)

       \(=\dfrac{100}{101}\)

\(c,K=\dfrac{4}{11.16}+\dfrac{4}{16.21}+\dfrac{4}{21.26}+...+\dfrac{4}{61.66}\)

        \(=\dfrac{4}{5}.\left(\dfrac{1}{11.16}+\dfrac{1}{16.21}+\dfrac{1}{21.26}+...+\dfrac{1}{61.66}\right)\)

        \(=\dfrac{4}{5}.\left(\dfrac{1}{11}-\dfrac{1}{16}+\dfrac{1}{16}-\dfrac{1}{21}+...+\dfrac{1}{61}-\dfrac{1}{66}\right)\)

        \(=\dfrac{4}{5}.\left(\dfrac{1}{11}-\dfrac{1}{66}\right)\)

       \(=\dfrac{4}{5}.\dfrac{5}{66}=>4.\dfrac{1}{66}\)

       \(=\dfrac{4}{66}=\dfrac{2}{33}\)

\(d,N=\dfrac{4}{1.3}+\dfrac{4}{3.5}+\dfrac{4}{5.7}+...+\dfrac{4}{99.101}\)

        \(=2.\left(\dfrac{1}{1.3}+\dfrac{1}{3.5}+\dfrac{1}{5.7}+...+\dfrac{1}{99.101}\right)\)

        \(=2.\left(1-\dfrac{1}{3}+\dfrac{1}{3}-\dfrac{1}{5}+...+\dfrac{1}{99}-\dfrac{1}{101}\right)\)

        \(=2.\left(1-\dfrac{1}{101}\right)\)

        \(=2.\dfrac{100}{101}\)

        \(=\dfrac{200}{101}\)

Bài 2:

\(K=\dfrac{5}{3.7}+\dfrac{5}{7.11}+\dfrac{5}{11.15}+...+\dfrac{5}{81.85}+\dfrac{5}{85.89}\)

    \(=\dfrac{5}{4}.\left(\dfrac{1}{3.7}+\dfrac{1}{7.11}+\dfrac{1}{11.15}+...+\dfrac{1}{81.85}+\dfrac{1}{85.89}\right)\)

    \(=\dfrac{5}{4}.\left(\dfrac{1}{3}-\dfrac{1}{7}+...+\dfrac{1}{85}-\dfrac{1}{89}\right)\)

    \(=\dfrac{5}{4}.\left(\dfrac{1}{3}-\dfrac{1}{89}\right)\)

    \(=\dfrac{5}{4}.\dfrac{86}{267}\)

    \(=\dfrac{215}{534}\)

Bài 3:

\(A=\dfrac{1}{25.24}+\dfrac{1}{24.23}+...+\dfrac{1}{7.6}+\dfrac{1}{6.5}\)

    \(=\dfrac{1}{5.6}+\dfrac{1}{6.7}+...+\dfrac{1}{23.24}+\dfrac{1}{24.25}\)

    \(=\dfrac{1}{5}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{7}+...+\dfrac{1}{24}-\dfrac{1}{25}\)

    \(=\dfrac{1}{5}-\dfrac{1}{25}\)

    \(=\dfrac{4}{25}\)

Bài 4 :

\(A=\dfrac{5}{3.6}+\dfrac{5}{6.9}+\dfrac{5}{9.12}+...+\dfrac{5}{99.102}\)

    \(=\dfrac{5}{3}.\left(\dfrac{1}{3.6}+\dfrac{1}{6.9}+\dfrac{1}{9.12}+...+\dfrac{1}{99.102}\right)\)

    \(=\dfrac{5}{3}.\left(\dfrac{1}{3}-\dfrac{1}{6}+\dfrac{1}{6}-\dfrac{1}{9}+...+\dfrac{1}{99}-\dfrac{1}{102}\right)\)

    \(=\dfrac{5}{3}.\left(\dfrac{1}{3}-\dfrac{1}{102}\right)\)

    \(=\dfrac{5}{3}.\dfrac{11}{34}\)

    \(=\dfrac{55}{102}\)

Bài 5 :

Sửa đề :\(a,E=\dfrac{1}{7}+\dfrac{1}{91}+\dfrac{1}{247}+\dfrac{1}{475}+\dfrac{1}{775}+\dfrac{1}{1147}\)

        \(=\dfrac{1}{1.7}+\dfrac{1}{7.13}+\dfrac{1}{13.19}+\dfrac{1}{19.25}+\dfrac{1}{25.31}+\dfrac{1}{31.37}\)

        \(=\dfrac{1}{6}.\left(\dfrac{1}{1.7}+\dfrac{1}{7.13}+\dfrac{1}{13.19}+\dfrac{1}{19.25}+\dfrac{1}{25.31}+\dfrac{1}{31.37}\right)\)

        \(=\dfrac{1}{6}.\left(1-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{13}+...+\dfrac{1}{31}-\dfrac{1}{37}\right)\)

        \(=\dfrac{1}{6}.\left(1-\dfrac{1}{37}\right)\)

        \(=\dfrac{1}{6}.\dfrac{36}{37}\)

        \(=\dfrac{6}{37}\)

\(b,C=\dfrac{2}{15}+\dfrac{2}{35}+\dfrac{2}{63}+\dfrac{2}{99}+\dfrac{2}{143}\)

       \(=\dfrac{2}{3.5}+\dfrac{2}{5.7}+\dfrac{2}{7.9}+\dfrac{2}{9.11}+\dfrac{2}{11.13}\)

       \(=\dfrac{1}{3}-\dfrac{1}{5}+\dfrac{1}{5}-\dfrac{1}{7}+...+\dfrac{1}{11}-\dfrac{1}{13}\)

       \(=\dfrac{1}{3}-\dfrac{1}{13}\)

       \(=\dfrac{10}{39}\)

Bài 6 :

\(a,\dfrac{3}{5.7}+\dfrac{3}{7.9}+\dfrac{3}{9.11}+...+\dfrac{3}{x\left(x+2\right)}=\dfrac{24}{35}\) 

    \(\dfrac{3}{2}\left(\dfrac{2}{5.7}+\dfrac{2}{7.9}+\dfrac{2}{9.11}+...+\dfrac{2}{x\left(x+2\right)}\right)=\dfrac{24}{35}\)

    \(\dfrac{3}{2}\left(\dfrac{1}{5}-\dfrac{1}{x+2}\right)=\dfrac{24}{35}\)

          \(\dfrac{1}{5}-\dfrac{1}{x+2}=\dfrac{24}{35}:\dfrac{3}{2}\)

          \(\dfrac{1}{5}-\dfrac{1}{x+2}=\dfrac{16}{35}\)

                  \(\dfrac{1}{x+2}=\dfrac{1}{5}-\dfrac{16}{35}\)

                  \(\dfrac{1}{x+2}=-\dfrac{9}{35}\)

                  \(-9\left(x+2\right)=1.35\)

                  \(-9\left(x+2\right)=35\)

                         \(x+2=35:-9\)

                        \(x+2=\dfrac{-35}{9}\)

                        \(x\)        \(=\dfrac{-35}{9}-2\)

                        \(x\)        \(=\dfrac{-53}{9}\)

Vậy \(x=\dfrac{-53}{9}\)

\(b,\dfrac{2}{4.7}+\dfrac{2}{7.10}+\dfrac{2}{10.13}+...+\dfrac{2}{x\left(x+3\right)}=\dfrac{1}{9}\)

   \(\dfrac{2}{3}.\left(\dfrac{3}{4.7}+\dfrac{3}{7.10}+\dfrac{3}{10.13}+...+\dfrac{3}{x\left(x+3\right)}\right)=\dfrac{1}{9}\)

   \(\dfrac{2}{3}.\left(\dfrac{1}{4}-\dfrac{1}{7}+\dfrac{1}{7}-\dfrac{1}{10}+...+\dfrac{1}{x}-\dfrac{1}{x+3}\right)=\dfrac{1}{9}\)

   \(\dfrac{2}{3}.\left(\dfrac{1}{4}-\dfrac{1}{x+3}\right)\)                                        \(=\dfrac{1}{9}\)

   \(\dfrac{1}{6}-\dfrac{2}{3.\left(x+3\right)}\)                                            \(=\dfrac{1}{9}\)

           \(\dfrac{2}{3.\left(x+3\right)}\)                                            \(=\dfrac{1}{6}-\dfrac{1}{9}\)

           \(\dfrac{2}{3.\left(x+3\right)}\)                                            \(=\dfrac{1}{18}\)

           \(\dfrac{2}{3.\left(x+3\right)}\)                                            \(=\dfrac{2}{36}\)

      ⇒   \(3.\left(x+3\right)=36\)

                 \(x+3=36:3\)

                 \(x+3=12\) 

                 \(x\)       \(=12-3\)

                 \(x\)       \(=9\)

Vậy \(x=9\)

Bài 7:

\(1+\dfrac{1}{3}+\dfrac{1}{6}+\dfrac{1}{10}+...+\dfrac{2}{x\left(x+1\right)}=1\dfrac{1989}{1991}\)

\(=>\dfrac{2}{2}+\dfrac{2}{6}+\dfrac{2}{12}+...+\dfrac{2}{x.\left(x+1\right)}=\dfrac{3980}{1991}\)

\(=>\dfrac{2}{1.2}+\dfrac{2}{2.3}+\dfrac{2}{3.4}+...+\dfrac{2}{x.\left(x+1\right)}=\dfrac{3980}{1991}\)

\(=>2.\left(\dfrac{1}{1.2}+\dfrac{1}{2.3}+\dfrac{1}{3.4}...+\dfrac{1}{x.\left(x+1\right)}\right)=\dfrac{3980}{1991}\)

\(=>2.\left(1-\dfrac{1}{2}+\dfrac{1}{2}-\dfrac{1}{3}+...+\dfrac{1}{x}-\dfrac{1}{x+1}\right)=\dfrac{3980}{1991}\)

\(=>2.\left(1-\dfrac{1}{x+1}\right)=\dfrac{3980}{1991}\)

             \(1-\dfrac{1}{x+1}=\dfrac{3980}{1991}:2\)

             \(1-\dfrac{1}{x+1}=\dfrac{1990}{1991}\)

                    \(\dfrac{1}{x+1}=1-\dfrac{1990}{1991}\)

                    \(\dfrac{1}{x+1}=\dfrac{1}{1991}\)

           \(=>x+1=1991\)

                  \(x\)       \(=1991-1\)

                  \(x\)       \(=1990\)

Vậy \(x=1990\)

1.

$2(x^2-4x+4)=(x-2)(x+3)$
$\Leftrightarrow 2(x-2)^2=(x-2)(x+3)$

$\Leftrightarrow 2(x-2)^2-(x-2)(x+3)=0$

$\Leftrightarrow (x-2)[2(x-2)-(x+3)]=0$

$\Leftrightarrow (x-2)(x-7)=0$

$\Leftrightarrow x-2=0$ hoặc $x-7=0$

$\Leftrightarrow x=2$ hoặc $x=7$

2.

$4x^2=9$

$\Leftrightarrow (2x)^2-3^2=0$

$\Leftrightarrow (2x-3)(2x+3)=0$

$\Leftrightarrow 2x-3=0$ hoặc $2x+3=0$

$\Leftrightarrow x=\frac{3}{2}$ hoặc $x=\frac{-3}{2}$
3.

$9x^2-1=(3x+1)(x+2)$

$\Leftrightarrow (3x+1)(3x-1)=(3x+1)(x+2)$

$\Leftrightarrow (3x+1)(3x-1)-(3x+1)(x+2)=0$

$\Leftrightarrow (3x+1)(3x-1-x-2)=0$

$\Leftrightarrow (3x+1)(2x-3)=0$

$\Leftrightarrow 3x+1=0$ hoặc $2x-3=0$

$\Leftrightarrow x=\frac{-1}{3}$ hoặc $x=\frac{3}{2}$

Các bạn giúp mình nha gợi ý : lưu ý trong 3 số tự nhiên liên tiếp có 1số chia hết cho 3