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5
∑
= 1
3
20
∑
=
1
12 + 12
20 2
Enriching the Lesson
Tell students the following legend: When the famous mathematician
Leonhard Euler was in grade school, the school master, in an
attempt to keep his students busy, told the students to find the sum of the
natural numbers from 1 to 100. The school master was quite disappointed
when Leonhard was able to produce the correct answer in about sixty seconds.
Show students the following method that was used by Euler. Then show
students how the formula S =
n
( a + a ) is a generalization of that method.
n
2 1
1 2 3 . . . . 98 99 100
+ 100 + 99 + 98 . . . . + 3 + 2 + 1
101 101 101 101 101 101 101 101 101 101
Sum = 101(100) ÷ 2 = 5,050
n
INSTRUCTION
Help students to learn summation
notation by translating the
symbols into English. For example,
5
∑
n= 1
3n
is read as “the sum of
the values of 3n from n = 1 to
n = 5.”
Have students revisit the
handshake problem which was
investigated in Lesson 9-1. Ask
students to write a summation
notation to represent the number
of handshakes.
Make sure that students do not
assume that the upper limit of a
summation notation is equivalent
to the number of terms that are
being summed. Advise students
that the index of a summation
notation is not always 1.
Make sure that students realize
that the formula S
n
n
= ( a + a
n)
2 1
does not apply to a summation
7
2
notation such as: ∑ n .
n=
1
Example 3 Reinforce to
students that the formula
S
n
n
= ( a + a
n) is just a
2 1
generalization of Euler’s method
which requires less work. Show
students how his method could
be used to determine the number
of seats in the auditorium.
9.2 Arithmetic Sequences and Series 399