Question

8. Use induction (on n) to show that: (a) (2n)! > 2" (n!)?, for n > 0. (w) (%) = (2+1), for os ms m. (0) Ž -»* (*) = (–1"("m"), for os m<n.

Answer 1

(a). (2n)! > 2" (n!!, for nyo For n=0, (2.0)! = 1 2010!) = 1 + (2n! ? 2"In he is true For no Suppose the statement is true 7or n=m lie., (2m)! 22,92131 - 113 For n = m+! (21m+1)! = 12m1! (2m+29124m+1) 22"? (myy. 2.6M+13 (27*1) [Using og = 2*! (maig'im+og+ [: 2m +1 2 m+1 = 2m+(muy mo the Statement is true or nam+1. Hence nor induction on as it is true for all nzo. 163 meter= (mm) for USMER For næmi So the statement is true n=m. Suppose the statment is true for n=2.i.e., any lehet Kam For n=2+1, i ml = 1 htt) + ( ) [using on ] so the statement is true for n=2+1 Hence

m induction on as it is true for all nem. (c). To E-04 ( h) = 4, "home for osman. For ma neno . - (7) = ( °(3) - | Holno") = 1 so the statement is true for m=0 Suppose the statement is true for mallie., A *) = (-7*("") - 139 For malt! en tot *() = **) + 1.20.) -1,#! styl maly + ( 23." [using ter] - 1,( 127.) - 16.) = Hit (n) So the statement is true for m=2+1. Hence by induction or r it is true for all osman.