Question

1) Cho A = 4 + 4^2 + 4^3 +4^4 + .....+ 4^30

Chứng minh A chia hết cho 5 , A chia hết cho 21

2) Cho B = 3^0 + 3^1 + 3^2 + 3^3 + 3^4 + ......+ 3^99

Chứng minh B chia hết cho 40

3) Cho A = 2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + .....+ 2^2016

Tìm số dư khi chia A cho 3

4) Cho S = 1 + 3 +3^2 + 3^3 + 3^4 + ...... +3^99 . Chứng tỏ : 2 . S + 1 là lũy thừa của 3

MỌI NGƯỜI GIÚP MK NHÉ

CẢM ƠN MỌI NGƯỜI

Answer 1

1) Ta có: \(A=4+4^2+4^3+4^4+...+4^{30}\)

\(=4\left(1+4\right)+4^3\left(1+4\right)+...+4^{29}\left(1+4\right)\)

\(=\left(1+4\right)\left(4+4^3+...+4^{29}\right)\)

\(=5\cdot\left(4+4^3+...+4^{29}\right)⋮5\)(đpcm)

Ta có: \(A=4+4^2+4^3+4^4+...+4^{30}\)

\(=4\left(1+4+4^2\right)+4^4\left(1+4+4^2\right)+...+4^{28}\left(1+4+4^2\right)\)

\(=\left(1+4+4^2\right)\left(1+4^4+...+4^{28}\right)\)

\(=21\cdot\left(1+4^4+...+4^{28}\right)⋮21\)(đpcm)

2) Ta có: \(B=3^0+3^1+3^2+3^3+3^4+...+3^{99}\)

\(=\left(3^0+3^1+3^2+3^3\right)+3^4\left(3^0+3^1+3^2+3^3\right)+...+3^{96}\left(3^0+3^1+3^2+3^3\right)\)

\(=40+3^4\cdot40+...+3^{96}\cdot40\)

\(=40\left(1+3^4+...+3^{96}\right)⋮40\)(đpcm)

4) Ta có: \(S=3^0+3^1+3^2+3^3+3^4+...+3^{99}\)

\(\Leftrightarrow3\cdot S=3+3^2+3^3+3^4+3^5+...+3^{100}\)

\(\Leftrightarrow3S-S=3+3^2+3^3+3^4+3^5+...+3^{100}-\left(3^0+3^1+3^2+3^3+3^4+...+3^{99}\right)\)

\(\Leftrightarrow2S=3+3^2+3^3+...+3^{100}-1-3-3^2-3^3-...-3^{99}\)

\(\Leftrightarrow2\cdot S=3^{100}-1\)

\(\Leftrightarrow2S+1=3^{100}\) là lũy thừa của 3(đpcm)