Question

Question 4 of the image

Prove that, for all n 1 1 Arrange the following rational numbers in increasing order: (i) x, is a rational number 61/99, 3/5, 17/30, 601/999, 599/1001. g 0 2 Find positive integers r and s such that r/s is equal to the repeating decimal (ii) 2 x5/2. Find an expression for x - 5 involving x,-5, and hence explain (without formal proof) why x, tends to a limit which is not a rational number 0.30024. [You need not give the answer in its 'lowest terms'. 5 It is known that v2 is an irrational number lying strictly between 1 and 2. Given a rational number x/y, construct an irra- tional number lying strictly between x/y and x/y + 10-10. You should explain briefly why the number you have constructed is irra 3 Find the repeating decimals for 1/13, 1/15, 1/17, 1/19. rs 4 The sequence {x,} is defined by the recursion (n 2). X -2, Xn-1 + Xn-1 l.

Answer 1

i2, n (2) Bymattamatical induatim Clearly fn n =1, e a. tet fer n K, NA ut fir nekt Xxt1 1 Put kk- ratigal na aner 32 1809462 clwrly a bounded seq. ut n (5-13 -C0 and Xn s boundeod Henee uppar bound nand Lo ue oound eg0 we get n convenses whch ielis that