Question

Cho S=2¹+2²+2³+...+2¹⁰⁰

a)Chứng minh S\(⋮\)15

b)Tìm chữ số tận cùng của S

c)Tính tổng S

 

Answer 1

a)\(S=2^1+2^2+...+2^{100}\)

\(=\left(2^1+2^2+2^3+2^4\right)+...+\left(2^{97}+2^{98}+2^{99}+2^{100}\right)\)

\(=2^1\left(1+2+2^2+2^3\right)+...+2^{97}\left(1+2+2^2+2^3\right)\)

\(=2^1\cdot15+...+2^{97}\cdot15\)

\(=15\cdot\left(2^1+...+2^{97}\right)⋮15\)

Answer 2

c)\(S=2^1+2^2+...+2^{100}\)

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

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

\(2S-S=\left(2^2+2^3+...+2^{101}\right)-\left(2+2^2+...+2^{100}\right)\)

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