\(S=1.\left(2-1\right)+2.\left(3-1\right)+...+100.\left(101-1\right)\)

   \(=1.2-1.1+2.3-1.2+...+100.101-1.100\)

   \(=\left(1.2+2.3+...+100.101\right)+\left(1+2+...+100\right)\)

   Áp dụng 1.2 + 2.3 + ... + n(n + 1) = \(\frac{n\left(n+1\right)\left(n+2\right)}{3}\) ta có 

\(S=\frac{100.101.102}{3}+\frac{100.101}{2}=343400+5050=\)348450

http://diendantoanhoc.net/topic/90149-1222321002/

a) A = 2 + 2² + 2³ + ... + 2¹⁰⁰

⇒ 2A = 2² + 2³ + 2⁴ + ... + 2¹⁰¹

⇒ A = 2A - A

= (2² + 2³ + 2⁴ + ... + 2¹⁰¹) - (2 + 2² + 2³ + ... + 2¹⁰⁰)

= 2¹⁰¹ - 2

b) B = 1 + 5 + 5² + ... + 5¹⁵⁰

⇒ 5B = 5 + 5² + 5³ + ... + 5¹⁵¹

⇒ 4B = 5B - B

= (5 + 5² + 5³ + ... + 5¹⁵¹) - (1 + 5 + 5² + ... + 5¹⁵⁰)

= 5¹⁵¹ - 1

⇒ B = (5¹⁵¹ - 1) : 4

A=2^101-1 chia chia cho 2

A = 2+2²+2³+2⁴+...+2¹⁰⁰

2A = 2² + 2³ + 2⁴ + 2⁵ +... + 2¹⁰¹

2A - A = ( 2² + 2³ + 2⁴ + 2⁵ +... + 2¹⁰¹ ) - ( 2+2²+2³+2⁴+...+2¹⁰⁰ )

A = 2¹⁰¹ - 2

A = 2+2²+2³+2⁴+...+2¹⁰⁰

2A = 2² + 2³ + 2⁴ + ... + 2¹⁰⁰

2A = 2² + 2³ + 2⁴ + ... + 2¹⁰¹

2A - A = ( 2² + 2³ + 2⁴ + ... + 2¹⁰¹ ) - ( 2² + 2³ + 2⁴ + ... + 2¹⁰⁰ )

1A = 2¹⁰¹ - 2

\(1A=\frac{2^{101}-2}{1}\)

a) Ta có: \(2A=2^2+2^3+2^4+2^5+2^6+...+2^{99}\) 

-

                \(A=2+2^2+2^3+2^4+2^5+2^6+...+2^{100}\)

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                \(A=2-2^{100}\)

Các bài khác cũng thế. Đây là mình tự nghĩ chứ không biết có đúng không. Có 60% sai! :) 

A = 1 + \(\frac{1}{2}\left(1+2\right)\)+ \(\frac{1}{3}\left(1+2+3\right)\)+ .... + \(\frac{1}{100}\left(1+2+3+...+100\right)\)

A = \(1+\frac{1}{2}\cdot\frac{2.3}{2}+\frac{1}{3}\cdot\frac{3.4}{2}+...+\frac{1}{100}\cdot\frac{100.101}{2}\)

A = \(\frac{2}{2}+\frac{3}{2}+\frac{4}{2}+...+\frac{101}{2}\)

A = \(\frac{2+3+4+...+101}{2}\)

A = \(\frac{\left(101+2\right).100}{2}\div2\)

A  = \(5150\div2=2575\)

A=1+3/2^3+4/2^4+5/2^5+...100/2^100
1/2*A = 1/2 + 3/2^4 + 4/2^5 +....+ 99/2^100 + 100/2^101

A- A/2 = 1/2A =1/2 + 3/2^3 + 1/2^4 +...+1/2^100 - 100/2^101=

= [1/2+1/2^2 +1/2^3 +...+1/2^100] -100/2^101 (Do 3/2^3 = 1/2^2 +1/2^3)

=[1-(1/2)^101]/(1-1/2) -100/2^101 =

=(2^101 -1)/2^100 - 100/2^101

=> A= (2^101 -1)/2^99 - 100/2^100