(i) Find a pair of positive integers, x1 and y1,that solve the equation

(x1)^2 -2 (y1)^2 = 1

(ii) Given integers a, bwe define two sequences x1,x2,x3and y1,y2,y3by setting

xn+1 = 3xn+ 4yn
yn+1 = axn+ byn

for n>=1

Find positive values for a, bsuch that

(xn+1)^2 -2 (yn+1)^2 = (xn)^2 -2 (yn)^2

(iii) Find a pair of integers X, Y which satisfy X^2 2Y^2 = 1 such that X > Y > 50

(iv) (Using the values of a and bfound in part (ii)) what is the approximate value of xn/yn as n increases?


I have done the question, and need help with the last bit.


i) I got x1 = 3 , y1= 4


ii) a=2, b=3


iii) X = 99, Y = 70


iv) i am not sure for this... I did xn/yn by taking xn-1 = yn-1 for n large, and that gives 7/5


but i dont think that is right, because if we approximate xn-1 = yn-, then xn = yn approximately, since they would get closer as they increase?