History of Hindu Mathematics: A Source Book v2.c17.s7.38

v2.c17.s7.38

In general, suppose a_r,¹, a_r,², a_r,r-1, a_r,r+1 a_r,n to be the amounts begged by the rth merchant from the others; and x_r the amount that he had initially. Then x_1 + \Sigma' a_{1,m} &= b_1 (\Sigma x - x_1 - \Sigma' a_{1,m}), / x_2 + \Sigma' a_{2,m} &= b_2 (\Sigma x - x_2 - \Sigma' a_{2,m}), / \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots / x_n + \Sigma' a_{n,m} &= b_n (\Sigma x - x_n - \Sigma' a_{n,m}), where Σ' a_r,m denotes summation from m = 1 to m = n excluding m = r. Therefore \Sigma x + (b_1 + 1)\Sigma' a_{1,m} &= (b_1 + 1)(\Sigma x - x_1), / \Sigma x + (b_2 + 1)\Sigma' a_{2,m} &= (b_2 + 1)(\Sigma x - x_2), / \dots\dots\dots\dots\dots\dots\dots\dots\dots\dots / \Sigma x + (b_n + 1)\Sigma' a_{n,m} &= (b_n + 1)(\Sigma x - x_n). Let k_r = (b_r + 1)Σ' a_r,m, r = 1, 2, 3,, n.