Question

[Cô Bé Yêu Đời]

Cho tổng \(M=2+2^2+2^3+2^4+...+2^{100}\)

a) Tổng M có chia hết cho 31 không ? Vì sao ?

b) Tính tổng M

Answer 1

[Nguyễn Huy Tú]

a) \(M=2+2^2+2^3+...+2^{100}\)

\(\Rightarrow M=\left(2+2^2+2^3+2^4+2^5\right)+...+\left(2^{96}+2^{97}+2^{98}+2^{99}+2^{100}\right)\)

\(\Rightarrow M=2\left(1+2+2^2+2^3+2^4\right)+...+2^{96}\left(1+2+2^2+2^3+2^4\right)\)

\(\Rightarrow M=2.31+...+2^{96}.31\)

\(\Rightarrow M=\left(2+...+2^{96}\right).31⋮31\)

\(\Rightarrow M⋮31\)

b) \(M=2+2^2+2^3+...+2^{100}\)

\(\Rightarrow2M=2^2+2^3+2^4+...+2^{101}\)

\(\Rightarrow2M-M=\left(2^2+2^3+2^4+...+2^{101}\right)-\left(2+2^2+2^3+...+2^{100}\right)\)

\(\Rightarrow M=2^{101}-2\)

Answer 2

[Nguyễn Đình Dũng]

a) M = 2 + 2² + 2³ + ... + 2¹⁰⁰

= (2+2²+2³+2⁴+2⁵) + (2⁶+2⁷+2⁸+2⁹+2¹⁰) + ... + (2⁹⁶+2⁹⁷+2⁹⁸+2⁹⁹+2¹⁰⁰)

= 2(1+2+2²+2³+2⁴) + 2⁶(1+2+2²+2³+2⁴) + ... + 2⁹⁶(1+2+2²+2³+2⁴)

= 31(2+2⁶+...+2⁹⁶) \(⋮\) 31

b) M = 2 + 2² + ... + 2¹⁰⁰

=> 2M = 2² + 2³ + ... + 2¹⁰¹

=> 2M - M = 2¹⁰¹ - 2

=> M = 2¹⁰¹ - 2