Question

tinh

A=1.2+2.3+3.4+....+N(N+1)

Answer 1

3A=1.2.3+2.3.3+3.4.3+...+n(n+1).3

3A=1.2(3-0)+2.3(4-1)+3.4(5-3)+....+n(n+1)(n-2)-(n-1)

3A=1.2.3-1.2.0+2.3.4-2.3.3+.+n(n+1)+(n+2)-(n-1)+n(n-1)

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

Answer 2

3A=1.2.3+2.3.3+3.4.3+.....+N(N+1).3

3A=1.2(3-0)+2.3(4-1)+3.4(5-3)+........+n(n+1)(n-2)-(n-1)

3a=1.2.3-1.2.0+2.3.4-2.3.3+....+n(n+1)+(n+2)-(n-1)+n(n+1)

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

Answer 3

3A=1.2.3+2.3.3+3.4.3+......+N(N+1).3

3A=1.2.(3-0)+2.3.(4-1)+3.4.(5-2)+....+N(N+1)(N-2)-(N-1)

3A=1.2.3-1.2.0+2.3.4-2.3.1+......+N(N+1)+(N+2)-N-1+N(N-1)

=>N(N-1)+(N+2)=N(N-1)-(N+2)/3

Answer 4

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

3A = 3[1.2 + 2.3 + 3.4 + ..... + n(n + 1)]

= 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)

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