Question

Evaluating Continued Fractions. Express each of the continued fractions given below as a simplified fraction in reduced form: (ii). [0,3,7,5,2 (iv). [-5,2,12]

Answer 1

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Remark: If continued fraction is given by [a_0, a_1, a_2, ...., a_n] (finite continued fraction) then, its fraction = a_0 + 1/a_1 + 1/a_2 + 1/a_3 ... + 1/a_n (i) [2, 4, 1, 5] = 2 + 1/4 + 1/1 + 1/5 = 2 + 1/4 + 1/6/5 = 2 + 1/4 + 5/6 = 2 + 1/29/6 = 2 + 6/29 = 64/29 (ii) [6, 1, 1] = 6 + 1/1 + 1/1 = 6 + 1/1 + 1 = 6 + 1/2 = 13/2 (ii) [0, 3, 7, 5, 2] = 0 + 1/3 + 1/7 + 1/5 + 1/2 = 0 + 1/3 + 1/7 + 1/11/2 = 0 + 1/3 + 1/7 + 2/11 = 0 + 1/3 + 11/79 = 0 + 79/248 = 79/248 (iv) [-5, 2, 1, 2] = -5 + 1/2 + 1/1 + 1/2 = -5 + 1/2 + 2/3 = -5 + 3/8 = -40 + 3/8 = -37/8