Sigma Notation
by CHED on June 17, 2017
lesson duration of 5 minutes
under Precalculus
generated on June 17, 2017 at 07:28 pm
Tags: Mathematical Induction
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Generated: Jun 18,2017 03:28 AM
Sigma Notation
( 5 mins )
Written By: CHED on July 2, 2016
Subjects: Precalculus
Tags: Mathematical Induction
Resources
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Content Standard
Key concepts of series and mathematical induction and the Binomial Theorem.
Performance Standard
Keenly observe and investigate patterns, and formulate appropriate mathematical statements and prove them using
mathematical induction and/or Binomial Theorem.
Learning Competencies
Use the sigma notation to represent a series
Writing and Evaluating Sums in Sigma Notation 1 mins
The sigma notation is a shorthand for writing sums. In this lesson, we will see the power of this notation in computing
sums of numbers as well as algebraic expressions.
Mathematicians use the sigma notation to denote a sum. The uppercase Greek letter ? (sigma) is used to indicate a
“sum.” The notation consists of several components or parts
Let f(i) be an expression involving an integer i. The expression
f(m) + f(m + 1) + f(m + 2) + · · · + f(n)
can be compactly written in sigma notation, and we write it as
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which is read “the summation of f(i) from i = m to n.” Here, m and n are integers with m ? n, f(i) is a term (or
summand) of the summation, and the letter i is the index, m the lower bound, and n the upper bound.
Example 2.2.1.
2.2.1. Expand each summation, and simplify if possible
Solution. We apply the definition of sigma notation.
Example 2.2.2. Write each expression in sigma notation.
(1) 1 + 1 2 + 1 3 + 1 4 + · · · + 1 100
(2) –1 + 2 – 3 + 4 – 5 + 6 – 7 + 8 – 9 + · · ·?25
(3) a2 + a4 + a6 + a8 + · · · + a20
(4) 1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128
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Solution.
The sigma notation of a sum expression is not necessarily unique. For example, the last item in the preceding example
can also be expressed in sigma notation as follows:
However, this last sigma notation is equivalent to the one given in the example.
Seatwork/Homework 1 mins
1. Expand each summation, and simplify if possible.
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2. Write each expression in sigma notation.
Teaching Notes
Equivalent answer: 1+3+5+· · ·+101 =
Properties of Sigma Notation 1 mins
We start with finding a formula for the sum of
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in terms of n.
The sum can be evaluated in different ways. A simple, though informal, approach is pictorial.
Teaching Notes
This illustration can be done with manipulatives, and allow the students to guess.
Another way is to use the formula for an arithmetic series with a1 = 1 and an=
S = n(a1 + an)/2 = n(n + 1)/2 .
We now derive some useful summation facts. They are based on the axioms of arithmetic addition and multiplication.
Proof.
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Proof.
Proof.
A special case of the above result which you might encounter more often is the following:
Proof.
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Note that the terms, f(m+1), f(m+2), . . . , f(n), all cancel out. Hence, we have
Example 2.2.3. Evaluate:
Solution.
Example 2.2.4. Evaluate:
1/1 · 2 + 1/2 · 3 + 1/3 · 4 + · · · + 1/99 · 100 .
Solution.
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Using f(i) = 1/i and the telescoping-sum property, we get
Example 2.2.5. Derive a formula for Xn i=1 i2 using a telescoping sum with terms f(i) = i3.
Solution. The telescoping sum property implies that
On the other hand, using expansion and the other properties of summation, we have
Equating the two results above, we obtain
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Finally, after dividing both sides of the equation by 6, we obtain the desired formula
Seatwork/Homework 1 mins
1. Use the properties of sigma notation to evaluate the following summations.
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Exercises 1 mins
1. Expand each sum.
2. Write each expression in sigma notation.
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Teaching Notes
Another possible answer for (b) is
3. Evaluate each sum.
.
4.
5.
6.
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7. Derive a formula for
by using a telescoping sum with terms f(i) = i4.
Answer: n2(n + 1)2/4
Download Teaching Guide Book 0 mins
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