Ta có : k(k+1)(k+2)-(k-1)(k+1)k

         =k(k+1).[(k+2)-(k-1)]

         =3k(k+1)

áp dụng  3(1+2)=1.2.3-0.1.2

             =>3(2.3)=2.3.4-1.2.3

             =>3(3.4)=3.4.5-2.3.4

            .....................................

              3n(n+1)=n(n+1)(n+2)-(n-1)n(n+1)

Cộng lại ta có   3.S=n(n+1)(n+2)=>S=n(n+1)(n+2)/3

CHÚC BẠN HỌC TỐT NHA !!!

k(k+1)(k+2)-(k-1)k(k+1)=k(k+1)(k+2-k+1)=3.k.(k+1)

S=1.2+2.3+3.4+...+n(n+1)

=>3S=1.2.3+2.3.3+3.4.3+...+n(n+1)3

=1.2.3+2.3.(4-1)+3.4(5-2)+...+n.(n+1)[(n+2)-(n-1)]

=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+n(n+1)(n+2)-(n-1)n(n+1)

=n(n+1)(n+2)

\(\Rightarrow S=\frac{n\left(n+1\right)\left(n+2\right)}{3}\)

m tưởng tao thik đăng à..............................................

\(s=\frac{n\left(n-1\right)\left(n+2\right)}{3}\)

Nha bạn

Chúc các bạn học giỏi

Ta có: k(k+1)(k+2)-(k-1)k(k+1)

        =k(k+1)[(k+2)-(k-1)]

        =k(k+1)[k+2-k+1]

        =k(k+1)[(k-k)+(2+1)]

        =k(k+1)3

        =3k(k+1)

 Vậy k(k+1)(k+2)-(k-1)k(k+1)=3k(k+1)

Áp dụng:

  S=1.2+2.3+3.4+...+n(n+1)

3S=3.1.2+3.2.3+3.3.4+...+3.n(n+1)

3S=1.2.3-0.1.2+2.3.4-1.2.3+3.4.5-2.3.4+...+n(n+1)(n+2)-(n-1)n(n+1)

3S=(1.2.3-1.2.3)+(2.3.4-2.3.4)+(3.4.5-3.4.5)+...+[(n-1)n(n+1)-(n-1)n(n+1)]+n(n+1)(n+2)-0

3S=n(n+1)(n+2)

  S=\(\frac{n\left(n+1\right)\left(n+2\right)}{3}\)

\(k\left(k+1\right)\left(k+2\right)-\left(k-1\right)k\left(k+1\right)=k\left(k+1\right)\left(k+2-k+1\right)=3\)\(\)\(k\left(k+1\right)\left(DPCM\right)\)

\(S=1.2+2.3+3.4+....+n\left(n+1\right)\)

\(3S=3\left[1.2+2.3+...+n\left(n+1\right)\right]\)

\(3S=1.2.3-0.1.2+2.3.4-1.2.3+....+n\left(n+1\right)\left(n+2\right)-\left(n-1\right)n\left(n+1\right)\)

\(3S=n\left(n+1\right)n\left(n+2\right)\)

\(S=\frac{n\left(n+1\right)\left(n+2\right)}{3}\)

Ta có:

k(k+1)(k+2)-(k-1)k(k+1)=k.(k+1).[(k+2)-(k-1)]

                                 =k.(k+1)(k+2-k+1)

                                 =3k.(k+1)

Phần 2 đề sai phải là tính S=1.2.3+2.3.4+...+n.(n+1).(n+2)

a) Xét trên tử

Ta có :

1.5.6 + 2.10.12 + 4.20.24 + 9.45.54

= 1.5.6 + \(^{2^3}\). 1.5.6 + \(^{4^3}\).1.5.6 + \(^{9^3}\).1.5.6

= 1.5.6 ( 2^3 + 4^3 + 9^3 )

Xét mẫu

Ta có :

1.3.5 + 2.6.10 + 4.12.20 + 9.27.45

= 1.3.5 + 2^3 .1.3.5 + 4^3 . 1.3.5 + 9^3 .1.3.5

= 1.3.5 ( 2^3 + 4^3 + 9^3 )

Ta có 

A = \(\frac{1.5.6.\left(2^3+4^3+9^3\right)}{1.3.5.\left(2^3+4^3+9^3\right)}\)= 2

b) Ta có :

 k(k+1)(k+2)-(k-1)k(k+1) = k(k + 1) (k + 2 - k + 1 ) = k( k + 1 ) . 3 = 3k( k + 1 )

Ta có :

S = 1.2 + 2.3 + 3.4 + ... + n(n + 1 )

\(\Rightarrow\)3S = 1.2.3 + 2.3.3 + 3.4.3 + ... + n(n + 1) . 3

3S = 1.2.3 + 2.3(4 - 1) + 3.4(5 - 2) + ... + n(n + 1)[(n + 2) - (n - 1)]

3S = 1.2.3 + 2.3.4 - 1.2.3 + 3.4.5 - 2.3.4 + 3.4.5 - 2.3.4 + ... + n(n + 1)(n + 2) - (n - 1)n(n + 1)

3S = n(n + 1)(n + 2)

S = \(\frac{n\left(n+1\right)\left(n+2\right)}{3}\)