MATH 3210 - SUMMER 2008 - ASSIGNMENT #2
Induction and
P
notation
(1) Expand the following expressions i.e. write the first 3 terms and last three terms
(with dots in between) in each sum:
n
X
(a)
i
i=0
(b)
n
X
(c)
n
X
(d)
n+1
X
(e)
n
X
3
(f)
i=0
n−2
X
(n − i)
i=0
2i
i=0
2k−1
k=1
4
2i
3 · 2j
j=−2
(2) Write the following expressions in
P
form:
(a) 1 + 3 + 5 + · · · + 13
(b) 6 + 9 + 12 + · · · + 24
(c) 1 + 3 + 9 + 27 + 81
(3) Find and prove formulas for the following expressions:
n+1
n
X
X
i−
j=
(a)
i=0
j=1
(b)
2n
X
2
(c)
n
X
1
(−3i − 2i) −
2n
X
i=0
k=1
n
X
(−3j 2 − 2j)
j=n
k
−
n
X
k=1
1
k+1
1
(d)
(hint: use the previous item).
k(k
+
1)
k=1
(4) Prove the following statements using induction (can you prove them in another way?)
(a) If 0 < a < b then for all n ∈ N: an < bn
1
(b) Prove that
n
X
(c) Prove that
n
X
2i = 2n+1 − 2
i=1
k=1
3
11
(7 + 3(k − 1)) = n2 + n
2
2
(d) Prove that for any real a, q and for any integer n:
n
(5) (a) Using the formula for nk prove that: nk = n−k
n
X
aq i =
i=0
n
k
(b) Prove the same thing but only using the definition of
number of k-element subsets of the set {1, . . . , n}).
(6) (a) Using the formula for nk prove that: n+1
= nk +
k
a − aq n+1
1−q
(i.e. that it is the
n
k−1
(b) Prove the same thing but only using the definition of
n
k
(i.e. that it is the
number of k-element subsets of the set {1, . . . , n}).
(7) Using induction prove the binomial formula (hint: if you get stuck you can use the
proof in page 12 of the text but you must explain every step).
2