1. By drfinition, (F pointa) (b) show that (n2+2n)N(2n+1) is O(s2) (3 pointi) (e) Prove thes (n+3)3=(n3)(3 points ) 2. (a) In 2+1=(22) whyld 15 points] (b) is 23=0(22]7 whe. (5 pointe) n,Z2,nisn,n3,len,nn3+7n3,m3+lon I(). ( 9 poriete) (a) T(n)=T(n1)+1/ (with T(0)=0 (bi)T(w)=T(n1)+en with T(0)=1, wheve e>1 las sear onostant (c) T( (n) )=2T(m1)+1 whh T(D)=1 (a) fc=fn1+ft2 with fu=0atelfc=1 (b) s3=fien1An1 uith m0=1 asel ug=0. recurrence a few bimes until a putiern enerpec. for instance, let us start with the nacumpoe T(n)=2T(n/2)+m. T(1) is (,1)= T(w)=2T(n/2)+cx=2(2T(n/z2)+n/2+rn=Z2T(n/x2)+PN==Z327(n/23)+cn(22)+3nn=Z2+T(n/Z2)+5m= wet T(m)=nT(1)+coslgn=A(wlgn). Do the name thiae ber the fullowing monereme T(n)=3T(2m)+m (a) What ie the prienal dth terme in thie coert is pointel) (b) What value of I shoubl be gunuged in to aet the aquevel is pointal (a) T(x)=2T(T)+n (b) T{(N)=3T(1)+n (c) T(w)=27T(3)+m3 (d) T(s)=5T(77)+m2 (a) T(n)=2T(n/4)+1 (b) T(m)=2T(n/i)+n (c) T(n)=2T(n/d)+n (d) T(n)=2T(n/4)+n2 1. By drfinition, (F pointa) (b) show that (n2+2n)N(2n+1) is O(s2) (3 pointi) (e) Prove thes (n+3)3=(n3)(3 points ) 2. (a) In 2+1=(22) whyld 15 points] (b) is 23=0(22]7 whe. (5 pointe) n,Z2,nisn,n3,len,nn3+7n3,m3+lon I(). ( 9 poriete) (a) T(n)=T(n1)+1/ (with T(0)=0 (bi)T(w)=T(n1)+en with T(0)=1, wheve e>1 las sear onostant (c) T( (n) )=2T(m1)+1 whh T(D)=1 (a) fc=fn1+ft2 with fu=0atelfc=1 (b) s3=fien1An1 uith m0=1 asel ug=0. recurrence a few bimes until a putiern enerpec. for instance, let us start with the nacumpoe T(n)=2T(n/2)+m. T(1) is (,1)= T(w)=2T(n/2)+cx=2(2T(n/z2)+n/2+rn=Z2T(n/x2)+PN==Z327(n/23)+cn(22)+3nn=Z2+T(n/Z2)+5m= wet T(m)=nT(1)+coslgn=A(wlgn). Do the name thiae ber the fullowing monereme T(n)=3T(2m)+m (a) What ie the prienal dth terme in thie coert is pointel) (b) What value of I shoubl be gunuged in to aet the aquevel is pointal (a) T(x)=2T(T)+n (b) T{(N)=3T(1)+n (c) T(w)=27T(3)+m3 (d) T(s)=5T(77)+m2 (a) T(n)=2T(n/4)+1 (b) T(m)=2T(n/i)+n (c) T(n)=2T(n/d)+n (d) T(n)=2T(n/4)+n2