Question

6. Use Mathematical Induction to show that (21 - 1)(2i+1) n for all integers n > 1. 2n +1 (5 marks) i=1

Answer 1

In this question we have to use the mathematical induction so for his we have to use the following steps:

Step 1: In this prove the given statement for n=1,2,3.

Sep 2: In this suppose the statement is true for n=k for some integer k $\geq$ 1

Then Step 3: By using the supposition in the Step 2 prove the statement for the n=k+1

Then we will have that this statement is true for all n $\geq$ 1

Mathematical Use Induction to show that coll Cat - Dla for all integrunn no is ky, made in cise supposition 31 (Qirolaiti) ant o have Cuswer! for Mathematical induction to prove following steps: 1,2,3 Step a Prove this for statement upto us enough true for Step-2 Let this statement Some integer Step 3. :) Then the To it is toue prone for RH Now always have some k integer such that The statement is proned true and thus we have true for RH and this RH becomes R and again for ht becomes so R is We have given statement is toue for all k and thus for all incegers moi Lets start! that Step 2 we 1,2, 3 then mul arbitraly Mence the Poore ! now Step 1: for n=1,2,3 Lus → (left hand side) Rns & Right hand side EWI 12-)) (aiti) n 2nt

2. n = 1 for sell Lns I to I g Şumatim cis fuom so islo oney { Q-1 (i-ai+ in I (2-) LaH) 211)-1) (211)+1) ㅗ 3 Rus n Je L 211)+1 e ym 2H 3 ant So Towe for n=1 for 2 Summation to a so LUS ) 2 in (i-1)(Qit1) + for i=2 for ich + (211) -1) (all) +1) (212)-1) (212)+1) + . (4-1)(4+1) (2-1) (2+1) ㅗ t ㅗ 3 Itt 3 15 315) sal う is la 15 a. 5 Auswer Rns m a > og jes Auswer 2nt ala) +1 Mene toue for n=2 also .

for 3 3 m3 Lus (21-vjeariti) in + (2015-1)201341) * (ata)13 (212+7 ㅗ (2la)-1(2(a) +) (213)-1)(30)+) + L + 그 (6-1)(6+1) (2-1)(24) 4-1)(4+1) & + 13315) (5)17) 3 (3) 15) (617) + (7) + (3) 35+ 713 (3) (5) (7) v (3)15) (7) 3 45 (3015)(7) 7 3 auswer Rns = ♡ 3 213) H 7 ant Also & True for n=3 NER.21 toul Step 2 Let it is for mo we have! R (21-1) (ait) 21 R Me - 70 È cat- QR H n=kt Step-3 To pecore for lie To Pecore ! RH RH ll RH 2(RH) +1 26+3 iz (21-1) (2iH)

Proof! 5 RH mow consider M (ai-1) (ait) 1에 + -1) (2 H) merim 21R+1)-0(ak+1)+1 R by using @ ie M (2-1)(Qih) R ARH 키 R. + ㅗ (@k+2-1) Car+241) a RH R f CƏRH) (@RH) (ak+3) R lak+3) +) 2 R²7 3241 (ARH) 2R+3) (art) (ak+3) > 28?art R + 1 V > ak (R+1) +1(+1) (2xH)[28+3) Car ) ( ar+3) JE (2RH) (R+1) ARH) (2R+3) Ru art 3 RA hence E RH lai-lai +1) art3 Henie True for m= RA $o It is tone for all n71 integers so Pooved by Mathematical Induction :