Question

Cho S = 1 + 3 + 3² + 3³ + … + 3⁴⁹

a) Chứng tỏ S chia hết cho 4.

b) Tìm chữ số tận cùng của S. 

Answer 1

a) \(S=1+3+3^2+3^3+...+3^{49}\)

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

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

\(=1.4+3^2.4+...+3^{48}.4\)

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

b) Ta có: \(S=1+3+3^2+3^3+...+3^{49}\)

\(3S=3+3^2+3^3+...+3^{49}+3^{50}\)

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

Ta thấy: \(3^{50}=3^{4.12}.3^2=\left(3^4\right)^{12}.3^2=81^{12}.9=...9\) (tận cùng là 9)

Suy ra \(3^{50}-1=\left(...9\right)-1=...8\) (tận cùng là 8)

Suy ra \(\Rightarrow S=\frac{3^{50}-1}{2}=\frac{\left(...8\right)}{2}=...4\Rightarrow S\) tận cùng là 4

Answer 2

a) \(S=1+3+3^2+3^3+...+3^{49}\)

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

\(S=4+\left(3^2.1+3^2.3\right)+....+\left(3^{48}.1+3^{48}.3\right)\)

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

\(S=1.4+3^2.4+...+3^{48}.4\)

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