Question

Prove the following statements: (a) If bn is recursively defined by bn = bn-1 + 3 for all integers n > 1 and bo = 2, then bn = 3n + 2 for all n > 0. (b) If an is recursively defined by cn = 3Cn-1 + 1 for all integers n > 1 and Co = 0, then cn = (3” – 1)/2 for all n > 0. (c) If dn is recursively defined by do = 1, dı = 4 and dn = 4dn–1 – 4dn-2 for all integers n > 2, then dn = (n + 1)2” for all n > 0.

PROVE BY INDUCTION

Answer 1

a) bn = bnr, +3,Wn21 p bo = 2 bu = 3n+2 , vnao >0 ..! 2, o =0 , We prove by induction on nyo If n=0, then bo = 3(0) +2 – 2 = 2 . 0 is true for n=o . Assume that is true for n=k i) bk >35+2. to prove o is true for n=k+1 6) to prove that bheti = 3(k+1) +2 i) to prove bete = 3k +5 Litlis = bikti = be + 3 F: bn = bn-,+3] E bk+3... = 34+3.43 [tey can induction assumption] = 3k +5 ; ; ; ; Ritis. .. . ... . .. . .: 0 is true for n=kti" llence by mathematical induction, Ö is true for all uso : :

(b) Cu =3Cn-, +1, Vn21, C =o Cu = (3-1) / no-> we prove @ is true for all uso If n=0, then co = (3-1) > or -soco Assume that is true for n=k te) Ce = (341% To pstore is true for n=kti ) to prove : Cami = (324_0) LHAS - CE cm= 3mmy+17 = 3 Eki), = 3 CK + = 3(3²-1) + = (364) + B = 4+1 3+2 [by our induction assumption] . K+ 34- 2 =R.His se io true for n=k+1. By the waflematical induction, @ all nyo. S te is true for