Mathematics in Ancient and Medieval India c7.s5.24

c7.s5.24

$p₁/q₁ = a₁$ $p₂/q₂ = a₁ + 1/a₂ = (a₁a₂ + 1)/a₂$ $p₃/q₃ = a₁ + 1/(a₂ +) 1/a₃$ $= a₁ + 1/(a₂ + 1/a₃)$ $= a₁ + a₃/(a₂a₃ + 1) = (a₁(a₂a₃ + 1) + a₃)/(a₂a₃ + 1)$ $p₄/q₄ = a₁ + 1/(a₂ +) 1/(a₃ +) 1/a₄$ $= a₁ + 1/(a₂ + 1/(a₃ + 1/a₄))$ $= a₁ + 1/(a₂ + a₄/(a₃a₄ + 1)) = a₁ + (a₃a₄ + 1)/(a₂(a₃a₄ + 1) + a₄)$ $= (a₁a₂(a₃a₄ + 1) + a₄ + a₃a₄ + 1)/(a₂(a₃a₄ + 1) + a₄)$ $and so on, and the last convergent p_n/q_n = a/b$