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If a1b1=a2b2=a3b3=⋯=anbn\dfrac{a_1}{b_1}=\dfrac{a_2}{b_2}=\dfrac{a_3}{b_3}=\dots =\dfrac{a_n}{b_n}b1a1=b2a2=b3a3=⋯=bnan then, each ratio equals m1a1+m2a2+m3a3+⋯+mnanm1b1+m2b2+m3b3+⋯+mnbn\dfrac{m_1a_1+m_2a_2+m_3a_3+\dots + m_na_n}{m_1b_1+m_2b_2+m_3b_3+\dots + m_nb_n}m1b1+m2b2+m3b3+⋯+mnbnm1a1+m2a2+m3a3+⋯+mnan for some constants mim_imi .
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