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Generating the Silver Ratio Using Different Approaches

Naresh Patel*,  C.L. Parihar**

*Department of Applied Sciences (Mathematics), Institute of Engineering & Technology, Devi Ahilya University, Indore-452017(M.P.) India.
Email: n_patel_1978@yahoo.co.in

**Indian Academy of Mathematics, 15-Kaushaliyapuri, Chitawad Road, Indore-452001 (M.P.) India Email: Profparihar@hotmail.com

Abstract:

Pell and Pell-Lucas numbers are respectively defined by the recurrence relation Their Binet formulae are and , where γ and δ                               are the roots of  x2 – 2x – 1 = 0 i.e. and so that or or . Also is called Silver ratio.

In this paper, we have generated Silver ratio using geometry, differential         equation, Newton-Raphson method, infinite continued fractions and bilinear   transformations. In the last, we have derived the formula for calculating number of digits in Pell and Pell-Lucas numbers, that is, and . Some of the interesting results of this paper are as follows:

(1) where Hence , it follows that the sequence of        approximations approaches as .

(2) Taking  a triangle in which the square of one side is four times the product of other two sides, then the square root of the ratio of two sides of such triangle lies between and .

(3) The infinite continued fraction converges to a limit (4) The Bilinear transformation has two fixed points and if and           only if , where a, b, c, d are integers; a, d > 0 and  ad-bc = 1.

(5) .

(6) Number of digits in = .

(7) Number of digits in = MSC 2000 Classification: 11B39

Key words: Silver Ratio, Pell and Pell-Lucas Numbers, Binet Formulae, Bilinear

Transformation, Number of Digits.

International eJournal of Mathematics and Engineering

Volume 2, Issue 2, Pages:  903 - 910