Assume-that-the-sequence-terms-tend-to-the-constant-value-u-so-that-as-n-u-n-1-u-and-u-n-u-i-show-that-u-2-u-1-0-ii-show-that-1-1-1-1-1-1-1-1-1-5-2-

Question Number 175211 by MathsFan last updated on 23/Aug/22

Assume that the sequence terms tend

to the constant value u, so that as

n \to \infty, u_{n - 1} \to u and u_{n} \to u .

(i) show that u^{2} + u - 1 = 0

(ii) show that \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \dots . .}}}} = \frac{- 1 + \sqrt{5}}{2}

Answered by Rasheed.Sindhi last updated on 23/Aug/22

(ii) u = \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \dots . .}}}}; u > 0

u = \frac{1}{1 + u}

u^{2} + u - 1 = 0

u = \frac{- 1 + \sqrt{1 - 4 (1) (- 1)}}{2} = \frac{- 1 + \sqrt{5}}{2}

[\frac{- 1 - \sqrt{5}}{2} < 0]

Commented by MathsFan last updated on 24/Aug/22

thank you sir