Ta có: \(A=1.3+2.4+3.5+4.6+...+99.101+100.102\)

\(A=1.\left(1+2\right)+2.\left(2+2\right)+3.\left(3+2\right)+4.\left(4+2\right)+....+99.\left(99+2\right)+100.\left(100+2\right)\)

\(A=\left(1^2+2^2+3^2+4^2+...+99^2+100^2\right)+\left(2+4+6+8+...+198+200\right)\)Đặt \(B=1^2+2^2+3^2+4^2+5^2+...+99^2+100^2\)

\(\Rightarrow B=\left(1^2+2^2+3^2+4^2+5^2+...+99^2+100^2\right)-2^2.\left(1^2+2^2+3^2+4^2+5^2+....+49^2+50^2\right)\)Tính dãy tổng quát \(C=1^2+2^2+3^2+4^2+5^2+...+n^2\)

\(C=1\left(0+1\right)+2\left(1+1\right)+3.\left(2+1\right)+4.\left(3+1\right)+5\left(4+1\right)+...+n\left[\left(n-1\right)+1\right]\)

\(C=\left[1.2+2.3+3.4+4.5+...+\left(n-1\right).n\right]+\left(1+2+3+4+5+....+n\right)\)

\(C=n.\left(n+1\right).\left[\left(n-1\right):3+1:2\right]=n.\left(n+1\right).\left(2n+1\right):6\)

Áp dụng vào B ta được:

\(B=100.101.201:6-4.50.51.101:6=166650\)

\(\Rightarrow A=166650+\left(200+2\right).100:2\)

\(\Rightarrow A=166650+10100=176750\)

Vậy A = 176750

Chúc bạn học tốt!!

=206                                 

206                                                                                                                              

\(4+5+89+12+8+88=9+101+96=110+96=206\)

\(\frac{87}{89}< \frac{1}{2}+\frac{1}{3\sqrt{2}}+...+\frac{1}{2011\sqrt{2010}}< \frac{88}{45}\)

Đặt \(A=\frac{1}{2}+\frac{1}{3\sqrt{2}}+...+\frac{1}{2011\sqrt{2010}}\)

\(\frac{1}{\sqrt{k}}-\frac{1}{\sqrt{k+1}}=\frac{1}{\sqrt{k\left(k+1\right)}}>\frac{1}{\left(k+1\right)\sqrt{k}}>\frac{1}{\left(k+1\right)k}=\frac{1}{k}-\frac{1}{k+1}\)

\(\Rightarrow1-\frac{1}{\sqrt{1}}+\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}}+...+\frac{1}{\sqrt{2010}}-\frac{1}{\sqrt{2011}}>A>1-\frac{1}{2}+\frac{1}{2}-\frac{1}{3}+...+\frac{1}{2010}-\frac{1}{2011}\)

\(\Rightarrow1-\frac{1}{\sqrt{2011}}>A>1-\frac{1}{2011}\)

\(\Rightarrow\frac{88}{45}>\frac{2011-\sqrt{2011}}{2011}>A>\frac{2010}{2011}>\frac{87}{89}\)

\(\Rightarrow\frac{87}{89}< \frac{1}{2}+\frac{1}{3\sqrt{2}}+...+\frac{1}{2011\sqrt{2010}}< \frac{88}{45}\)

Có 90 số hạng ,chia thành 45 nhóm,mỗi nhóm có 2 số hạng

B=(1+6)+(6^2+6^3)+...+(6^86+6^87)+(6^88+6^89)

B=7+(6^2x1+6^3x6)+...+(6^86x1+6^86x6)+(6^88x1+6^88x6)

B=7+6^2x(1+6)+..+6^86.7+6^88x7+

B=7+6^2x7+...+6^86x7+6^88x17

B=7x(6^2+..+6^86+6^88)

Vậy B:3

Có 90 số ,chia thành 30 nhóm mỗi nhóm có 3 số hạng

B=(1+6+6^2)+(6^3+6^4+6^5)+...+(6^88+6^89+6^90)

B=43x1+6^3x(1+6+36)+.....+6^88x(1+6+36)

Còn lại tự làm