1
Mathematical induction
1.1
A conjecture about Sums of odd numbers
Let’s recall that the
ith
Odd number is given by the expression
2i1.
So we can make a table whose
columns are labeled
by
i
, and show the odd nu mbers and their sums.
I
1
2
3
4
5
6
Odd number
1
3
5
7
9
Running sum
1
4
9
16
25
We notice that the running sums (that is, adding the next odd number to the previous sum) seem to always
give us a perfect square. In fact, looking more closely, it seems to give us exactly the square of the number
i,
that is used to calculate the
ith
odd number. This might lead us to
conjecture
that there is an underlying
rule that is always true.
We have to do a couple of things now. First, we have to figure out how to state that rule mathematically.
Second, we have to see how a mathematician would go about proving the rule. We can not say that it would
be checked for every value of
i,
because that would take an infinite amount of time, and this course ends in
May.
So, first to express this mathematically. We need an expression for the sum of the first “i” odd numbers.
But since we are also using “i” in the formula for the odd numbers, we are going to have to use another
letter “n” to indicate how many odd numbers we are adding up.
We were happy to discover that everyone in the class has seen the use of the capital Greek letter Sigma to
represent a sum [although sum of us may have forgotten exactly what it means.]
The expression for the sum of the first “N” odd numbers is given by:
1
( )
(2
1)
iN
i
Sum N
i
Note that the N appears only once on the right hand side, as the value of the “upper limit”. We can
pronounce this expression as: “The sum, from i=1 to i=N, of the quantity (2i1).”. In
this expression the
“i=1” is called the “lower limit”. The “i=N” is called the upper limit; and the “(2i1)” is called the
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 Fall '09
 Boros
 Addition, Mathematical Induction, Natural number

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