HW 2 Solution . Note : In the solution we only sketch the proofs or provide the ideas of { nets When you write proofs of induction in HW or exam , you should ) try your best to write in more detail and tollogwjhnentenfelztafneesa ? 5 . I : th E INT , 31 h3 - h Pf : For A- I , 31 o ✓ . Suppose true for n=k , suppose 3 / K ' - K Note that CktD3 - ( Kt D= Ks -13k¥ 3k -11 - K - I = k3 - Kt 3 ( Rtk ) - 31 K ' K . So 31 Ktp -441 ) By CMI ) , we have th E INT , 31 h3 - h Ah 5.4 : Prove for th EIN , tx # , tox ' = ' II ! Pf : For n - - O , Xo = I = 1¥ ✓ Suppose there for A- K II :X ' = xhtxk " = ' II + xktl = I xktltxktt ( t - X ) I - X
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HW 2 Solution .
Note : In the solution we only sketch the proofs or provide the ideas of{nets . When you write proofs of induction in HW or exam , you should)try your best to write in more detail and tollogwjhnentenfelztafneesa? 5 . I : th E INT , 31 h3 - h .
Pf : For A- I , 31 o . .
Suppose true for n=k , suppose 3 / K ' - K
.
Note that CktD3 - (KtD= Ks -13k¥3k -11 - K- I
= k3 -Kt 3(Rtk) -
By CMI) , we have th E INT , 31 h3 - h .
Ah
5.4 : Prove for th EIN , tx# , tox' = 'II! Pf : For n -- O ,
Xo = I = 1¥ .
II:X' = xhtxk" = 'II + xktl
I - xktz =
I - x
an
5. 6 : fete HEIN , define { Uo=0Uktl =3Uk -13k 1570
Prove that Un -_ n 3h - t for th EIN .
Pf : For n=o , Uo=o V .
Suppose thee for h=k ,
UKH = 34k -13k =3 . K 3k" t 3k .
= 3k ( KH) .
Pf : For h=I , 3/9 V .
Suppose the for h=K .
They 4kt ' -15=4 (4kt 5) - 15 .
Note that 314458 3/15 .
.
Pf : For n=4 , 4 ! = 24716=24 .
Suppose true n=k , then (KH) ! = Cktt ) . K ! > Cktl) - 2K 7 2. zK=zkH
Done by induction .
Pfc . For n -
IIe , Letty = Ea Ican t attunes
= # t k¥47 =
Done by induction .
.
So 9kt , = 6%4,52 =
T @k<5) .
Note : In the solution we only sketch the proofs or provide the ideas of{nets . When you write proofs of induction in HW or exam , you should)try your best to write in more detail and tollogwjhnentenfelztafneesa? 5 . I : th E INT , 31 h3 - h .
Pf : For A- I , 31 o . .
Suppose true for n=k , suppose 3 / K ' - K
.
Note that CktD3 - (KtD= Ks -13k¥3k -11 - K- I
= k3 -Kt 3(Rtk) -
By CMI) , we have th E INT , 31 h3 - h .
Ah
5.4 : Prove for th EIN , tx# , tox' = 'II! Pf : For n -- O ,
Xo = I = 1¥ .
II:X' = xhtxk" = 'II + xktl
I - xktz =
I - x
an
5. 6 : fete HEIN , define { Uo=0Uktl =3Uk -13k 1570
Prove that Un -_ n 3h - t for th EIN .
Pf : For n=o , Uo=o V .
Suppose thee for h=k ,
UKH = 34k -13k =3 . K 3k" t 3k .
= 3k ( KH) .
Pf : For h=I , 3/9 V .
Suppose the for h=K .
They 4kt ' -15=4 (4kt 5) - 15 .
Note that 314458 3/15 .
.
Pf : For n=4 , 4 ! = 24716=24 .
Suppose true n=k , then (KH) ! = Cktt ) . K ! > Cktl) - 2K 7 2. zK=zkH
Done by induction .
Pfc . For n -
IIe , Letty = Ea Ican t attunes
= # t k¥47 =
Done by induction .
.
So 9kt , = 6%4,52 =
T @k<5) .
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