Question 6
Let $a_1,a_2,\ldots,a_n$ be positive numbers and $q$ a given real number, $0\lt q\lt 1$. Find $n$ real numbers $b_1,b_2,\ldots,b_n$ that satisfy:
(1) $a_k\lt b_k$ for all $k=1,2,\ldots,n$;
(2) $q\lt \frac{b_{k+1}}{b_k}\lt \frac1q$ for all $k=1,2,\ldots,n-1$;
(3) $b_1+b_2+\cdots+b_n\lt \frac{1+q}{1-q}(a_1+a_2+\cdots+a_n)$.
(1) $a_k\lt b_k$ for all $k=1,2,\ldots,n$;
(2) $q\lt \frac{b_{k+1}}{b_k}\lt \frac1q$ for all $k=1,2,\ldots,n-1$;
(3) $b_1+b_2+\cdots+b_n\lt \frac{1+q}{1-q}(a_1+a_2+\cdots+a_n)$.