Question

Bài 7. Cho S = 1 + 2 + 2^2 + 2^3 + ..... 2^20.So sánh với 5.2^19

Answer 1

S=1+2+2^2+2^3+...+2^20

2.S=2+2^2+2^3+...+2^20+2^21

2.S-S=S=(2+2^2+2^3+....+2^21)-(1+2+2^2+...+2^20)

S=2^21-1

bây giờ so sánh 2^21-1 với 5.2^19

mà 2^21-1=2^19.2^2-1 hay 2^19 .4 -1 <2^19.5

=>S<2^19.5

Answer 2

\(S=1+2+2^2+2^3+\cdots+2^{20}\)

=> \(2S=2+2^2+2^3+2^4+\cdots+2^{21}\)

=> \(2S-S=\left(2+2^2+2^3+2^4+\cdots+2^{21}\right)-\left(1+2+2^2+2^3+\ldots+2^{20}\right)\)

=>\(S=2^{21}-1\)

Mà \(2^{21}-1=2^{19}.2^2-1\) hay \(2^{19}.4-1<2^{19}.5\)

=>\(S<2^{19}.5\)

Answer 3

Ta có: \(S=1+2+2^2+\cdots+2^{20}\)

=>\(2S=2+2^2+2^3+\cdots+2^{21}\)

=>\(2S-S=2+2^2+\cdots+2^{21}-1-2-\cdots-2^{20}\)

=>\(S=2^{21}-1\)

\(2^{21}-1-5\cdot2^{19}=2^{19}\left(2^2-5\right)-1=-2^{19}-1<0\)

=>\(S-5\cdot2^{19}<0\)

=>\(S<5\cdot2^{19}\)