Question

1.\(A=7+7^3+7^5+.....+7^{1999}\)

Chứng minh: A \(⋮\) 35

2. Cho \(S=1+3+3^2+3^3+....+3^{48}+3^{49}\)
a, Chứng tỏ S\(⋮\) 4
b, Tìm chữ số tận cùng của S
c, Chứng tỏ \(S=\frac{3^{50}-1}{2}\)

Answer 1

Bài 1:

\(A=7+7^3+7^5+...+7^{1999}\)

\(\Rightarrow A=\left(7+7^3\right)+\left(7^5+7^7\right)+...+\left(7^{1997}+7^{1999}\right)\)

\(\Rightarrow A=\left(7+343\right)+7^4\left(7+7^3\right)+...+7^{1996}\left(7+7^3\right)\)

\(\Rightarrow A=350+7^4.350+...+7^{1996}.350\)

\(\Rightarrow A=\left(1+7^4+...+7^{1996}\right).350⋮35\)

\(\Rightarrow A⋮35\left(đpcm\right)\)

b2:

a) \(S=1+3+3^2+...+3^{49}\)

\(\Rightarrow S=\left(1+3\right)+\left(3^2+3^3\right)+...+\left(3^{48}+3^{49}\right)\)

\(\Rightarrow S=\left(1+3\right)+3^2\left(1+3\right)+...+3^{48}\left(1+3\right)\)

\(\Rightarrow S=4+3^2.4+...+3^{48}.4\)

\(\Rightarrow S=\left(1+3^2+...+3^{48}\right).4⋮4\)

\(\Rightarrow S⋮4\left(đpcm\right)\)

c) \(S=1+3+3^2+...+3^{49}\)

\(\Rightarrow3S=3+3^2+3^3+...+3^{50}\)

\(\Rightarrow3S-S=\left(3+3^2+3^3+...+3^{50}\right)-\left(1+3+3^2+...+3^{49}\right)\)

\(\Rightarrow2S=3^{50}-1\)

\(\Rightarrow S=\frac{3^{50}-1}{2}\left(đpcm\right)\)

Answer 2

\(A=7+7^3+7^5+...+7^{1999}\)

\(=\left(7+7^3\right)+\left(7^5+7^7\right)+...+\left(7^{1997}+7^{1999}\right)\)

\(=\left(7+7^3\right)+7^4\left(7+7^3\right)+...+7^{1996}\left(7+7^3\right)\)

\(=350+7^4.350+...+7^{1996}.350\)

\(=350.\left(1+7^4+...+7^{1996}\right)\)

Vì \(350⋮35\) nên \(350.\left(7+7^4+...+7^{1996}\right)⋮35\)

Vậy \(A⋮35\)(đpcm)

2.

a, \(S=1+3+3^2+...+3^{49}\)

\(=\left(1+3\right)+\left(3^2+3^3\right)+...+\left(3^{48}+2^{49}\right)\)

\(=\left(1+3\right)+3^2\left(1+3\right)+...+3^{48}\left(1+3\right)\)

\(=4+3^2.4+...+3^{48}.4\)

\(=4\left(1+3^2+...+3^{48}\right)\)

Vì \(4⋮4\) nên \(4\left(1+3^2+...+3^{48}\right)⋮4\)

Vậy \(S⋮4\) (đpcm)

c, \(S=1+3+3^2+...+3^{49}\)

\(3S=3+3^2+3^3+...+3^{50}\)

\(3S-S=\left(3+3^2+3^3+...+3^{50}\right)-\left(1+3+3^2+...+3^{49}\right)\)

\(2S=3^{50}-1\)

\(S=\left(3^{50}-1\right):2\) (đpcm)

b, Ta có: \(S=\left(3^{50}-1\right):2\)

\(=\left(3^{48}.3^2-1\right):2\)

\(=\left[\left(3^4\right)^{14}.9-1\right]:2\)

\(=\left[\overline{\left(...1\right)}^{14}.9-1\right]:2\)

\(=\left[\overline{\left(...1\right)}.9-1\right]:2\)

\(=\left[\overline{\left(...9\right)}-1\right]:2\)

\(=\overline{\left(...8\right)}:2\)

\(=\overline{...4}\)

Vậy chữ số tận cùng của S là 4