MATH1061/7861, Thu 27 Aug 2020
00 78255
All
equal
Git
14k e Cd
me
Mr
Gies
j
(b) Use your change of variable to simplify the following expression:
why7 Charge of vaiabh.gg
i E 3Cj42j
i
czje3 Ig
Question 2. Let k ∈ Z, k ≥ 2. Which of the following expressions are equal?
Eiai E
a
equal b
ia a
d Gi1 4k
t
m
o 2jtD an
f
ii
hal
a aiD1 261112k
et n f ii
Question 1.
(a) What expression do you obtain when you apply the change of variable j = i + 1 to the following sum?
i
2cg Dts
E 2513 q zit 3
E
E
iE 91,2 n I
pi
3J 18J 13
e
IE
f
i
to
B P fshow pg Ime
reduction
want
prove predicater k’t
Let in ring g n En
Base case
show
PCD PG pts Pla 7
g Things
7 TM
My
Question 3. Use induction to prove that, for each integer n ∈ N: Te
na
LHS Rtl s
Equal
Pf
Ute
Gi D induction
Basis step in 4 Need to prove
Inducthest
Need to Deduce
EBeiste
lies
K
K2 21kt
holdsK2 12kt I KH 2 RHS
we
know
A
i
i By induction i Zi l in Vnc N.tt
zit f 2llzll
Rl
i holds
Assume 62k ngphoYuesisPCk Gi D gl
Ealzi D Kei
Pfktc
RHS
GiD EGiD t.EEfao
LHS
s
I Tyincketche hypothesis
GiD re Aet proof
n
It
smaffutes
Thirdproot
EGi D
n Intl Question 4. Prove that, for each integer n > 1, 5n + 9 < 6n.
Range of n 2,3 4,5
Proof Use induction.TN
n n n n in g
Basis step
n Z
LHS 549259 RHS e 62 36
V holds
L 6k where k 72 66kt to
34
Inductive step
Assume 549
Deduce 5k11 9
24 ncn
tl
21 tn z.TTT
36 Basis step
34C
1
i
InD Gi D
Et Htt
n
35
9 14 46s
Challenge A. Prove that, ∀x ∈ N, x5/5 + x3/3 + 7x/15 ∈ Z.
Challenge B. Which values can you make using only 3¢ and/or 5¢ stamps? Prove your answer correct!