Question

[Nguyễn Thanh Thành Đạt]

Bài 1:

Cho A=2+2^2+2^3+…+2^60

a) Thu gọn tổng A

b) Chứng tỏ rằng: A chia hết cho 3,5,7

Bài 2:

Cho C=5+5^2+5^3+...+5^20.Chứng minh rằng:

a) C chia hết cho 5

b) C chia hết cho 6

c) C chia hết cho 13

Bài 3:

Cho C=1+3^1+3^2+3^3+...+3^11

a) Rút gọn C

b) Chứng minh C chia hết cho 10

Giải nhanh giúp mình nha!(các bước rõ ràng)

Answer 1

[Nguyễn Lê Phước Thịnh]

Bài 1:

b:

\(A=2+2^2+2^3+...+2^{60}\)

\(=\left(2+2^2+2^3\right)+\left(2^4+2^5+2^6\right)+...+\left(2^{58}+2^{59}+2^{60}\right)\)

\(=2\left(1+2+2^2\right)+2^4\left(1+2+2^2\right)+...+2^{58}\left(1+2+2^2\right)\)

\(=7\cdot\left(2+2^4+...+2^{58}\right)⋮7\)

 \(A=2+2^2+2^3+...+2^{60}\)

\(=\left(2+2^2+2^3+2^4\right)+\left(2^5+2^6+2^7+2^8\right)+...+\left(2^{57}+2^{58}+2^{59}+2^{60}\right)\)

\(=2\left(1+2+2^2+2^3\right)+2^5\left(1+2+2^2+2^3\right)+...+2^{57}\left(1+2+2^2+2^3\right)\)

\(=15\left(2+2^5+...+2^{57}\right)=3\cdot5\cdot\left(2+2^5+...+2^{57}\right)\)

=>\(A⋮3;A⋮5\)

Bài 2:

a: \(C=5+5^2+...+5^{20}\)

\(=5\left(1+5+...+5^{19}\right)⋮5\)

b: \(C=5+5^2+5^3+...+5^{20}\)

\(=\left(5+5^2\right)+\left(5^3+5^4\right)+...+\left(5^{19}+5^{20}\right)\)

\(=5\left(1+5\right)+5^3\left(1+5\right)+...+5^{19}\left(1+5\right)\)

\(=6\left(5+5^3+...+5^{19}\right)⋮6\)

c: \(C=5+5^2+5^3+...+5^{20}\)

\(=\left(5+5^2+5^3+5^4\right)+\left(5^5+5^6+5^7+5^8\right)+...+\left(5^{17}+5^{18}+5^{19}+5^{20}\right)\)

\(=5\left(1+5+5^2+5^3\right)+5^5\left(1+5+5^2+5^3\right)+...+5^{17}\left(1+5+5^2+5^3\right)\)

\(=156\left(1+5^5+...+5^{17}\right)\)

\(=12\cdot13\cdot\left(1+5^5+...+5^{17}\right)⋮13\)

Bài 3:

a: \(C=1+3+3^2+...+3^{11}\)

=>\(3C=3+3^2+3^3+...+3^{12}\)

=>\(3C-C=3+3^2+...+3^{12}-1-3-...-3^{11}\)

=>\(2C=3^{12}-1\)

=>\(C=\dfrac{3^{12}-1}{2}\)

b: \(C=1+3+3^2+3^3+...+3^{11}\)

\(=\left(1+3+3^2+3^3\right)+\left(3^4+3^5+3^6+3^7\right)+\left(3^8+3^9+3^{10}+3^{11}\right)\)

\(=\left(1+3+3^2+3^3\right)+3^4\left(1+3+3^2+3^3\right)+3^8\left(1+3+3^2+3^3\right)\)

\(=40\left(1+3^4+3^8\right)=4\cdot10\cdot\left(1+3^4+3^8\right)⋮10\)

Answer 2

[Lê Trần Xuân Phúc]

Ta có: 2A = 22 + 23 + … + 2⁶⁰

2A – A = (22 + 23 + … + 2⁶⁰) – (2 + 22 + 23 + … + 2⁵⁹)

A = 2⁶⁰  – 2.