For question number 2 on the first sheet, I used proof by induction. Does my proof look correct? Or would you have solved it differently?The piece that is throwing me off is where the -7(3^k)+7(3^k). I understand they cancel each other and it helps solve the proof, but don’t understand where they come from. as follows4C1) - 3, 4C2)=うforヘ234 divides 7^ - 3^ forandProvethatしeとScritleWorkeachnEN INote: Prook by inductionフ-31Pf:Letヘ37-3a set of -naturatinumbers, 's vch7^-3^ is avisible by 4 for eachヘこ2レ-) 72-32 4thatレ4nEN.BS)When n=), then7-3' =4 and .5 oivaible by4. Therefore, 16S.Letへe 5IH)For n=k, then 7k -3k isdiusible by 4The, weneed to prove k+lESis Oivisible by 4 ILet nンkt1, show 7kい-3k0l7kリー3ck)- プko)フ(3)+フし3)-3(m).(3- ר)3 ~ ("3 - יח)ר ב= 7(4*)- 34(4)(k-1)Now,Sinee b.th sides are being multiples bythis implies Hey are bth di visibleK+)by 4. Thactue, 7-3*l is owistble by4 and by inducthio, this proves 7":3* isfor esch nENdarisible by 4 1) Prove that 2" < n! for any integer n > 42) Prove that 4 divides 7" -3" for eachn E N3) Prove that sin nx| < n| sin x| for all n EN and x E R.4) Define f : N → R as follows:f(1) = 3, f(2) =3and for n > 3 f(n) =f(n- 1)+ f(n- 2)2'Prove that f(n) = 2+ (-)"-1 for all n eN1十….十41S2 Vn EN5) Prove that 1+x + y6) Let x > 0 and y > 0, prove that Vry <

Name: For question number 2 on the first sheet, I used proof by induction. D
Uploaded: 2024-03-20T07:07:26.000Z
Channel: Bartleby
Description: For question number 2 on the first sheet, I used proof by induction. Does my proof look correct? Or would you have solved it differently?The piece that is throwing me off is where the -7(3^k)+7(3^k). I understand they cancel each other and it helps s