1/

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

\(=\left(1.2+2.3+3.4+...+99.100\right)-\left(1+2+3+...+99\right)=\)

Đặt 

\(A=1.2+2.3+3.4+...+99.100\)

\(3A=1.2.3+2.3.3+3.4.3+...+99.100.3=\)

\(=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+99.100.\left(101-98\right)=\)

\(=1.2.3-1.2.3+2.3.4-2.3.4+3.4.5-...-98.99.100+99.100.101=\)

\(=99.100.101\Rightarrow A=\dfrac{99.100.101}{3}=33.100.101\)

Đặt

\(B=1+2+3+...+99=\dfrac{99.\left(1+99\right)}{2}=4950\)

\(\Rightarrow N=A-B\)

2/

Số hạng cuối cùng là 10000 hoặc 1000000 mới làm được

\(A=1^2+2^2+3^2+...+100^2\) 

Tính như câu 1

3/ Làm như bài 4

4/

\(S=1^2+3^2+5^2+...+99^2=\)

\(=1.\left(3-2\right)+3\left(5-2\right)+5\left(7-2\right)+...+99\left(101-2\right)=\)

\(=\left(1.3+3.5+5.7+...+99.101\right)-2\left(1+3+5+...+99\right)\)

Đặt

\(B=1+3+5+...+99=\dfrac{50.\left(1+99\right)}{2}=2500\) 

Đặt

\(A=1.3+3.5+5.7+...+99.101\)

\(6A=1.3.6+3.5.6+3.7.6+...+99.101.6=\)

\(=1.3.\left(5+1\right)+3.5.\left(7-1\right)+5.7.\left(9-3\right)+...+99.101.\left(103-97\right)=\)

\(=1.3+1.3.5-1.3.5+3.5.7-3.5.7+5.7.9-...-97.99.101+99.101.103=\)

\(=3+99.101.103\Rightarrow A=\dfrac{3+99.101.103}{6}\)

\(\Rightarrow S=A-2B\)

Bài 1:

\(N=1^2+2^2+3^3+...+99^2\)

\(N=1.1+2.2+3.3+...+99.99\)

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

\(N=1.2-1+2.3-2+3.4-3+...+99.100-99\)

\(N=\left(1.2+2.3+3.4+...+99.100\right)-\left(1+2+3+...+99\right)\)

Đặt \(\left\{{}\begin{matrix}A=1.2+2.3+3.4+...+99.100\\B=1+2+3+...+99\end{matrix}\right.\)

+) Tính \(A=1.2+2.3+3.4+...+99.100\)

Ta có:

\(3A=1.2.3+2.3.3+3.4.3+...+99.100.3\)

\(3A=1.2.3+2.3.\left(4-1\right)+3.4.\left(5-2\right)+...+99.100.\left(101-98\right)\)

\(3A=1.2.3+2.3.4-1.2.3+3.4.5-2.3.4+...+99.100.101-98.99.100\)

\(3A=99.100.101\)

\(\Rightarrow A=\dfrac{99.100.101}{3}=333300\)

+) Tính \(B=1+2+3+...+99\)

\(B\) có số số hạng là: \(\dfrac{99-1}{1}\) + 1 = 99 (số hạng)

\(\Rightarrow B=\dfrac{\left(99+1\right).99}{2}=4950\)

\(\Rightarrow N=A-B=333300-4950=328350\)

\(\Rightarrow N=328350\)

xin loi mik danh nham nhe bai do la 10000 nhe

2.2-2.2=0

(2.2):(2.2)=1

2:2+2:2=2

2.2-2:2=3

(2.2.2):2=4

Đúng chưa hở

2 + 2 - 2 - 2 = 0

2 : 2 + 2 - 2 = 1

2 : 2 + 2 : 2 = 2

2 × 2 - 2 : 2 = 3

2 × 2 - 2 + 2 = 4

2+2-2-2=0

2:2+2-2=2

2+2-2:2=3

2+2+2-2=4

học tốt^^

a) \(S=1+2+2^2+...+2^{100}\)

\(2S=2+2^2+2^3+...+2^{101}\)

\(2S-S=\left(2+2^2+...+2^{101}\right)-\left(1+2+...+2^{100}\right)\)

\(S=2^{101}-1\)

b) \(X=2^{2012}-2^{2011}-...-2-1\)

\(X=2^{2012}-\left(1+2+...+2^{2011}\right)\)

Đặt \(X=2^{2012}-Y\)

Ta có :

\(Y=1+2+...+2^{2011}\)

\(2Y=2+2^2+...+2^{2012}\)

\(2Y-Y=\left(2+2^2+...+2^{2012}\right)-\left(1+2+...+2^{2011}\right)\)

\(Y=2^{2012}-1\)

\(\Rightarrow X=2^{2012}-2^{2012}+1\)

\(\Rightarrow X=1\)

\(\Rightarrow2010X=2010\)

c=1+2+4+8+64

=79

\(C=2^0+2^1+2^2+2^3+2^0+2^1+2^2+2^3\)

\(=\left(2^0.2\right)+\left(2^1.2\right)+\left(2^2.2\right)+\left(2^3.2\right)\)

\(=2+4+8+16\)

\(=\left(2+8\right)+\left(4+16\right)\)

\(=10+20\)

\(=30\)

\(D=\left(2^0+2^1+2^2+2^3\right).2^0.2^1.2^2.2^3\)

\(=\left(1+2+4+8\right).1.2.4.8\)

\(=\left(8+2+4+1\right).1.2.4.8\)

\(=\left(10+4+1\right).1.2.4.8\)

\(=15.1.2.4.8\)

\(=\left(15.2\right).1.4.8\)

\(=30.1.4.8\)

\(=120.8\)

\(=960\)

=(2^78+79+80):(2^77+76+75)

=2²³⁷:2²²⁸

=2^237-228

=2⁹

=512

Học Sinh Giỏi À bạn ?

Bạn Thi Học Sinh Giởi À 

1/đặt M= 1-2 = -1

      N= 2^2-2^3+...-2^2005+2^2006

Ta có

N= 2^2-2^3+...-2^2005+2^2006    

2N=2^3-2^4+...-2^2006+2^2007

2N-N=(2^3-2^4+...-2^2006+2^2007)-(2^2-2^3+...-2^2005+2^2006)

N=2^2007-2^2

=) S=M+N

      = -1+2^2007-2^2

=)2S= -2+2^2008-2^3

       = 2^2008-10

=)3S= -3+3.2^2007-3.2^2

       = 3.2^2007-15

2/ =)3S-2^2007=3.2^2007-15 -2^2007

        =2.2^2007-15

Vậy 1/ 2S= 2^2008-10

          3S= 3.2^2007-15

      2/ 3S-2^2007= 2.2^2007-15

ta có 1² - 2² = - 3

       3² - 4² = - 7

      .................

    2005² - 2006² =    -4011

-{(4011+3)[(4011-3):4+1]:2} = -2013021 

\(x\) \(\times\) \(\dfrac{1}{4}\) = 6 : 1 : 2

\(x\) \(\times\) \(\dfrac{1}{4}\) = 6:2

\(x\) \(\times\) \(\dfrac{1}{4}\) =  3

\(x\)         = 3 : \(\dfrac{1}{4}\)

\(x\)        = 12

Mình nhầm đoạn 6:1/2.nhờ bạn giải lại hộ mình với

\(x\)\(\times\) \(\dfrac{1}{4}\) = 6 : \(\dfrac{1}{2}\)

\(x\) \(\times\) \(\dfrac{1}{4}\) = 6 \(\times\) \(\dfrac{2}{1}\)

\(x\) \(\times\) \(\dfrac{1}{4}\) = 12

\(x\)       =  12 : \(\dfrac{1}{4}\)

\(x\)      = 48

Tính nhanh:

                         A =         \(\dfrac{2}{3}\) + \(\dfrac{2}{9}\) + \(\dfrac{2}{27}\) + \(\dfrac{2}{81}\) + \(\dfrac{2}{243}\) + \(\dfrac{2}{729}\)

                  A \(\times\) 3 =  2  +  \(\dfrac{2}{3}\) + \(\dfrac{2}{9}\) + \(\dfrac{2}{27}\) + \(\dfrac{2}{81}\) + \(\dfrac{2}{243}\) 

            A \(\times\) 3 - A = 2 - \(\dfrac{2}{729}\)

           A \(\times\) ( 3 - 1) = 2 \(\times\) ( 1 - \(\dfrac{1}{729}\))

             A \(\times\) 2       = 2 \(\times\) \(\dfrac{728}{729}\)

            A = 2 \(\times\) \(\dfrac{728}{729}\) : 2

           A = \(\dfrac{728}{729}\)

c, tính:

 \(\dfrac{4}{5}\) + \(\dfrac{1}{2}\) - \(\dfrac{1}{6}\)

= \(\dfrac{4}{5}\) + \(\dfrac{1}{3}\)

= \(\dfrac{17}{15}\)