Math 100
Induction and Sums
Martin H. Weissman
1.
Some Sums
Suppose that
a
0
,a
1
,a
2
,...
is a sequence of real numbers. Then, one may create a new
sequence, by summation: for all natural numbers
n
, deﬁne:
s
n
=
n
X
i
=0
a
i
.
One could deﬁne the sequence
s
n
recursively as following:
•
The ﬁrst term is deﬁned by
s
0
=
a
0
.
•
For all
n
∈
N
, if
n >
0, then
s
n
=
a
n
+
s
n

1
.
First, we will experiment with summation. Given the following sequences, determine the
sequence of sums:
(1) 1
,
2
,
4
,
8
,
16
,
32
,...
(2) 1
,
3
,
5
,
7
,
9
,
11
,...
(3) 0
,
1
,
2
,
3
,
4
,
5
,
6
,...
(4) 1
/
2
,
1
/
6
,
1
/
12
,
1
/
20
,...
(5) 1
,
1
/
2
,
1
/
4
,
1
/
8
,
1
/
16
,...
Can you guess formulas for the sequences of sums? Today, we’ll spend some time proving
these formulas. After guesses are made, we’ll prove the guesses, using the following proof as
a template.
Theorem 1.
Let
a
0
,a
1
,a
2
,...
be the sequence, which satisﬁes
a
n
= 2
n
for all natural numbers
n
. Let
s
n
=
∑
n
i
=0
a