Rút gọn biểu thức B = sin^3 a/3 + 3sin^3 a/3^2 + 3^2.sin^3 a/3^3

* Hướng dẫn giải

Đáp án B

$$\begin{gathered} B={\sin }^3\frac{a}{3}+3{\sin }^3\frac{a}{3^2}+3^2{\sin }^3\frac{a}{3^3}+\mathrm{...}+3^{n-1}{\sin }^3\frac{a}{3^n} \\ =\frac{3\sin \frac{a}{3}-\sin a}{4}+3.\frac{3\sin \frac{a}{3^2}-\sin \frac{a}{3}}{4}+3^2.\frac{3\sin \frac{a}{3^3}-\sin \frac{a}{3^2}}{4}+\mathrm{...}+3^{n-1}.\frac{3\sin \frac{a}{3^n}-\sin \frac{a}{3^{n-1}}}{4} \\ =\frac{1}{4}.\left(-\sin a+3\sin \frac{a}{3}-3\sin \frac{a}{3}+3^2\sin \frac{a}{3^2}-3^2\sin \frac{a}{3^2}+3^3\sin \frac{a}{3^3}-\mathrm{...}-3^{n-1}\sin \frac{a}{3^{n-1}}+3^n\sin \frac{a}{3^n}\right) \\ =\frac{1}{4}\left(3^n\sin \frac{a}{3^n}-\sin a\right) \\ =\frac{3^{\mathrm{n}}\sin \frac{\mathrm{a}}{3^{\mathrm{n}}}-\mathrm{sina}}{4} \end{gathered}$$