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