mat3166 list.pdf - MAT 3166 1. Show that for all n ≥ 1 n X...

MAT 3166 Problems + Assignment 1
1. Show that for all n ≥ 1,
(i)
n
X
k =1
k
3
=
n
2
( n + 1)
2
4
, (ii)
n
X
k =1
k ( k !) = ( n + 1)! − 1, (iii)
n
X
k =0
2
k
n
k
= 3
n
.
2. Let ( a
n
)
n ≥ 0
a sequence defined recursively by
a
0
= 0 , a
1
= 1 , et a
n
= 3 a
n − 1
− 2 a
n − 2
if n ≥ 2 .
(i) Complute a
0
, . . . , a
5
. Conjecture a formula for a
n
and prove it by induction.
(ii) Let F =
3 − 2
1 0
. Show that for all n ≥ 1, we have
a
n +1
a
n
a
n
a
n − 1
= F
n − 1
3 1
1 0
.
3. Show that for every choice of integers a and b with b ̸ = 0, there exist integers q and r
with a = qb + r and −| b | / 2 < r ≤ | b | / 2. Is this expression unique?
4. Let m and n be positive integers and r the remainder of the Euclidian division of m by
n . Show that the remainder of the division of 2 m − 1 by 2 n − 1 is 2 r − 1.
5. Let m and n be positive integers and let d = gcd( m, n ). Using the results of the previous
problem, show that gcd(2 m − 1 , 2 n − 1) = 2 d − 1.
6. In each case, determine the gcd of a