Question

Cho A=\(3+3^2+3^3+...+3^{30}\)

a) Chứng minh rằng A\(\vdots\)13 và A \(\vdots\)52

b) Hỏi A có phải là số chính phương không? Tại sao?

Answer 1

1)

a) \(A=3+3^2+3^3+3^4+3^5+3^6+....+3^{28}+3^{29}+3^{30}\)

\(\Leftrightarrow A=\left(3+3^2+3^3\right)+\left(3^4+3^5+3^6\right)+....+\left(3^{28}+3^{29}+3^{30}\right)\)

\(\Leftrightarrow A=3\left(1+3+3^2\right)+3^4\left(1+3+3^2\right)+....+3^{28}\left(1+3+3^2\right)\)

\(\Leftrightarrow A=3.13+3^4.13+....+3^{28}.13\)

\(\Leftrightarrow A=13\left(3+3^4+....+3^{28}\right)⋮13\left(dpcm\right)\)

b) \(A=3+3^2+3^3+3^4+3^5+3^6+....+3^{25}+3^{26}+3^{27}+3^{28}+3^{29}+3^{30}\)

\(\Leftrightarrow A=\left(3+3^2+3^3+3^4+3^5+3^6\right)+....+\left(3^{25}+3^{26}+3^{27}+3^{28}+3^{29}+3^{30}\right)\)

\(\Leftrightarrow A=3\left(1+3+3^2+3^3+3^4+3^5\right)+....+3^{25}\left(1+3+3^2+3^3+3^4+3^5\right)\)

\(\Leftrightarrow A=3.364+....+3^{25}.364\)

\(\Leftrightarrow A=364\left(3+3^5+3^{10}+....+3^{25}\right)\)

\(\Leftrightarrow A=52.7\left(3+3^5+3^{10}+....+3^{25}\right)⋮52\left(dpcm\right)\)

Answer 2

2) \(A=3+3^2+3^3+....+3^{30}\)

\(\Leftrightarrow3A=3\left(3+3^2+3^3+....+3^{30}\right)\)

\(\Leftrightarrow3A=3^2+3^3+3^4+....+3^{30}+3^{31}\)

\(\Leftrightarrow3A-A=\left(3^2+3^3+3^4+....+3^{30}+3^{31}\right)-\left(3+3^2+3^3+....+3^{30}\right)\)

\(\Leftrightarrow2A=3^{31}-3\)

\(\Leftrightarrow A=\dfrac{3^{31}-3}{2}\)

Vậy A không phải là số chính phương