Question

Cho : M = 1 + 3 + 3² + 3³ + .... + 3⁹⁹⁹

a) CMR : M chiaa hết cho 4

b) Tìm số dư khi chia cho 13

c) Tính gọn M

giúp mình nha các bạn

 

Answer 1

a) Ta có:

\(M=1+3+3^2+3^3+...+3^{999}\)

\(M=\left(1+3\right)+\left(3^2+3^3\right)+...+\left(3^{998}+3^{999}\right)\)

\(M=4+3^2.\left(1+3\right)+...+3^{998}.\left(1+3\right)\)

\(M=4+3^2.4+...+3^{998}.4\)

\(M=\left(1+3^2+...+3^{998}\right).4⋮4\)

\(\Rightarrow M⋮4\)

Answer 2

c) \(M=1+3+3^2+...+3^{999}\)

\(\Rightarrow3M=3+3^2+3^3+...+3^{1000}\)

\(\Rightarrow3M-M=\left(3+3^2+3^3+...+3^{1000}\right)-\left(1+3+3^2+3^3+...+3^{999}\right)\)

\(\Rightarrow2M=3^{1000}-1\)

\(\Rightarrow M=\frac{3^{1000}-1}{2}\)

Answer 3

b) \(M=1+3+3^2+3^3+...+3^{999}\)

\(M=\left(1+3+3^2\right)+\left(3^3+3^4+3^5\right)+...+\left(3^{997}+3^{998}+3^{999}\right)\)

\(M=15+3^3.\left(1+3+3^2\right)+...+3^{997}.\left(1+3+3^2\right)\)

\(M=15+3^3.15+...+3^{997}.15\)

\(M=\left(1+3^3+...+3^{997}\right).15\)

Vì 15 chia 13 có số dư là 2 nên M chia 13 có số dư là 2

Vậy M chia 13 dư 2