Question

[Nguyễn Đức Anh]

Tính

\(B=1^2+2^2+3^2+4^2+5^2+...+98^2\)

Answer 1

[Truong_tien_phuong]

Ta có \(B=1^2+2^2+3^2+.....+98^2\)

\(\Rightarrow B=1.1+2.2+3.3+.....+98.98\)

\(\Rightarrow B=1.\left(2-1\right)+2.\left(3-1\right)+.....+98.\left(99-1\right)\)
\(\Rightarrow B=\left(1.2-1\right)+\left(2.3-2\right)+.....+\left(98.99-98\right)\)

\(\Rightarrow B=\left(1.2+2.3+....+98.99\right)-\left(1+2+....+98\right)\)

\(\Rightarrow B=\left(1.2+2.3+...+98.99\right)-\left(\frac{\left(1+98\right).98}{2}\right)\)

\(\Rightarrow B=\left(1.2+2.3+...+98+99\right)-\left(4851\right)\)⁽¹⁾ 

Đặt \(M=1.2+2.3+....+98.99\)

\(\Rightarrow3M=1.2.3+2.3.3+......+98.99.3\)

\(\Rightarrow3M=1.2.\left(3+0\right)+2.3.\left(4-1\right)+.....+98.99.\left(100-97\right)\)

\(\Rightarrow3M=\left(1.2.3+0.1.2\right)+\left(2.3.4-1.2.3\right)+.......+\left(98.99.100-97.98.99\right)\)

\(\Rightarrow3M=98.99.100-0.1.2\)

\(\Rightarrow3M=970200-0\)

\(\Rightarrow3M=970200\)

\(\Rightarrow M=\frac{970200}{3}\)

\(\Rightarrow M=323400\)

\(\Rightarrow1.2+2.3+....+98.99=323400\)

Thay \(1.2+2.3+......+98.99\) \(=323400\) Vào (1) ta được:

\(B=323400-4851\)

\(\Rightarrow B=318549\)

Vậy \(B=318549\)

Answer 2

[Mạnh Lê]

\(B=1.1+2.2+3.3+4.4+5.5+...+98.98\)\(=1.\left(2-1\right)+2.\left(3-1\right)+...+98.\left(99-1\right)\)

\(\left(1.2+2.3+3.4+...+98.99\right)-\left(1+2+...+99\right)\)

\(\Rightarrow a-b=\left(1.2+2.3+...+98.99\right)\)\(-\left[\left(1.2+2.3+...+98.99\right)-\left(1+2+...+98\right)\right]\)\(=1+2+3+...+98\)

Tổng của dãy số trên là : \(a-b=\frac{\left(98+1\right).98}{2}=4851\).