thu gọn tổng sau: A= 2 + 2^2 + 2^3 +2^4 + ... + 2^99 + 2^100
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![](https://rs.olm.vn/images/avt/0.png?1311)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(A=2+2^2+2^3+2^4+...+2^{99}+2^{100}\)
\(\Rightarrow2A=2^2+2^3+2^4+...+2^{100}+2^{101}\)
\(\Rightarrow A=2A-A=2^2+2^3+2^4+...+2^{100}+2^{101}-2-2^2-2^3-2^4-...-2^{99}-2^{100}=2^{101}-2\)
![](https://rs.olm.vn/images/avt/0.png?1311)
a, Ta có :
A = 1 + 2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100
2A = 2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100 + 2 101
A = 2A – A = ( 2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100 + 2 101 ) –( 1 + 2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100 )
= 2 + 2 2 + 2 3 + 2 4 + . . . + 2 99 + 2 100 + 2 101 – 1 - 2 - 2 2 - 2 3 - 2 4 - . . . - 2 99 - 2 100
= 2 101 - 1
Vậy A = 2 101 - 1
b, Ta có.
B = 5 + 5 3 + 5 5 + . . . + 5 97 + 5 99
5 2 B = 5 2 ( 5 + 5 3 + 5 5 + . . . + 5 97 + 5 99 )
25B = 5 3 + 5 5 + . . . + 5 97 + 5 99 + 5 101
25B – B = ( 5 3 + 5 5 + . . . + 5 97 + 5 99 + 5 101 ) – ( 5 + 5 3 + 5 5 + . . . + 5 97 + 5 99 )
24B = 5 3 + 5 5 + . . . + 5 97 + 5 99 + 5 101 – 5 - 5 3 - 5 5 - . . . - 5 97 - 5 99
24B = 5 101 - 5
B = 5 101 - 5 24 = 5 5 100 - 1 24
Vậy B = 5 5 100 - 1 24
![](https://rs.olm.vn/images/avt/0.png?1311)
Có : \(S=1+2+2^2+2^3+....+2^{99}\)
\(\Rightarrow2S=2+2^2+2^3+....+2^{100}\)
\(\Rightarrow2S-S=\left(2+2^2+2^3+...+2^{100}\right)-\left(1+2+2^2+....+2^{99}\right)\)
\(\Rightarrow S=2^{100}-1< 2^{100}\)
Vậy \(S< 2^{100}\)
S=1+2+22+23+....+299
⇒2S=2+22+23+....+2100
⇒2S−S=2100-1
S=2100-1
vì 2100 -1<2100
⇒S<2100
![](https://rs.olm.vn/images/avt/0.png?1311)
\(2^{100}-2^{99}+2^{98}-2^{97}+2^{96}-2^{95}+...+2^4-2^3+2^2\)
\(=\left(2^{100}-2^{99}+2^{98}\right)-\left(2^{97}-2^{96}+2^{95}\right)+...+\left(2^4-2^3+2^2\right)\)
\(=2^{96}\left(2^4-2^3+2^2\right)-2^{93}\left(2^4-2^3+2^2\right)+...+\left(2^4-2^3+2^2\right)\)
\(=12\left(2^{96}-2^{93}+...+1\right)⋮12\)
![](https://rs.olm.vn/images/avt/0.png?1311)
Ta có
2 1 + 2 2 + 2 3 + 2 4 + 2 5 + 2 6 + 2 7 +...+ 2 98 + 2 99 + 2 100
= 2 1 + ( 2 2 + 2 3 + 2 4 ) + ( 2 5 + 2 6 + 2 7 ) +...+ ( 2 98 + 2 99 + 2 100 )
= 2 + 2 2 1 + 2 + 2 2 + 2 5 1 + 2 + 2 2 + . . . + 2 98 1 + 2 + 2 2
= 2 + 2 2 . 7 + 2 5 . 7 + . . . + 2 98 . 7 = 2 + 7 2 2 + 2 5 + . . . + 2 98
Mà 7 . 2 2 + 2 5 + . . . + 2 98 ⋮ 7
Nên 2 + 7 2 2 + 2 5 + . . . + 2 98 : 7 d ư 2
\(A=2+2^2+...+2^{99}+2^{100}\)
\(2A=2^2+2^3+...+2^{101}\)
\(2A-A=\left(2^2+2^3+....+2^{101}\right)-\left(2+2^2+...+2^{100}\right)\)
\(A=2^{101}-2\)
A= 2+2^2+2^3+...+2^99+2^100
=>2A=2^2+2^3+2^4+...+2^100+2^101
=> 2A - A =(2^2+2^3+2^4+...+2^100+2^101)-(2+2^2+2^3+...+2^99+2^100)
=>A = 2^101-2