Question

Cho B= 3+ 3²+3³+3⁴+...+3³⁰

^{a) Tính B?}

^{b)Chứng tỏ B chia hết cho4?}

Answer 1

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

\(\Rightarrow3B=3^2+3^3+3^4+...+3^{31}\)

\(\Rightarrow3B-B=\left(3^2+3^3+3^4+...+3^{31}\right)-\left(3+3^2+3^3+....+3^{30}\right)\)

\(\Rightarrow2B=3^{31}-3\)

\(\Rightarrow B=\frac{3^{31}-3}{2}\)

b) \(B=3+3^2+3^3+3^4+...+3^{30}\)

         \(=\left(3+3^2\right)+\left(3^3+3^4\right)+....+\left(3^{29}+3^{30}\right)\)

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

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

Vậy B chia hết cho 4