Question

tìm số hạng đầu của cấp số nhân tăng (\(u_n\)) có \(u_1u_2u_3=4096\); và \(S_3=56\)

Answer 1

\(\left\{{}\begin{matrix}u_1.u_1.q.u_1.q^2=4096\\u_1.\dfrac{q^3-1}{q-1}=56\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}\left(u_1q\right)^3=4096\\u_1\left(q^2+q+1\right)=56\end{matrix}\right.\)

\(\left\{{}\begin{matrix}u_1q=16\\u_1\left(q^2+q+1\right)=56\end{matrix}\right.\)

\(\Rightarrow\dfrac{16}{q}\left(q^2+q+1\right)=56\)

\(\Leftrightarrow16q^2-40q+16=0\Rightarrow\left[{}\begin{matrix}q=1\\q=\dfrac{1}{2}\left(loại\right)\end{matrix}\right.\)

\(\Rightarrow u_1=8\)

Answer 2

\(\left\{{}\begin{matrix}u_1u_1u_1qq^2=4096\\u_1+u_1q+u_1q^2=56\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}u_1^3q^3=4096\\u_1\left(1+q+q^2\right)=56\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}u_1q=16\\u_1\left(1+q+q^2\right)=56\end{matrix}\right.\)

\(\Rightarrow\dfrac{1+q+q^2}{q}=\dfrac{7}{2}\Leftrightarrow2+2q+2q^2=7q\Rightarrow\left[{}\begin{matrix}q=2\\q=\dfrac{1}{2}\left(loai\right)\end{matrix}\right.\)

\(\Rightarrow u_1=\dfrac{16}{2}=8\)